Ms Excel Moving Average Analysis

Eva Goldwater Centro de Consultoria em Bioestatística Escola de Saúde Pública da Universidade de Massachusetts atualizada em fevereiro de 2007 Em resumo Utilizamos o Excel para fazer algumas tarefas básicas de análise de dados para ver se é uma alternativa razoável ao uso de um pacote estatístico para as mesmas tarefas. Concluímos que o Excel é uma má escolha para análise estatística além dos exemplos de livros didáticos, as estatísticas descritivas mais simples ou para mais do que algumas poucas colunas. Os problemas que encontramos que levaram a esta conclusão estão em quatro áreas gerais: Valores em falta são tratados de forma inconsistente, e às vezes incorretamente. Organização de dados difere de acordo com a análise, forçando você a reorganizar seus dados de muitas maneiras, se você quiser fazer muitas análises diferentes. Muitas análises só podem ser feitas em uma coluna de cada vez, tornando inconveniente fazer a mesma análise em muitas colunas. A produção é mal organizada, às vezes inadequadamente rotulada, e não há registro de como uma análise foi realizada. Excel é conveniente para a entrada de dados, e para manipular rapidamente linhas e colunas antes da análise estatística. No entanto, quando você estiver pronto para fazer a análise estatística, recomendamos o uso de um pacote estatístico, como SAS, SPSS, Stata, Systat ou Minitab. Introdução Excel é provavelmente a planilha mais comumente usada para PCs. Computadores recém-comprados freqüentemente chegam com o Excel já carregado. É facilmente usado para fazer uma variedade de cálculos, inclui uma coleção de funções estatísticas, e um Data Analysis ToolPak. Como resultado, se de repente você achar que você precisa fazer alguma análise estatística, você pode recorrer a ele como a escolha óbvia. Decidimos fazer alguns testes para ver como o Excel serviria como um aplicativo de análise de dados. Para apresentar os resultados, usaremos um pequeno exemplo. Os dados para este exemplo são fictícios. Optou-se por ter duas variáveis ​​categóricas e duas contínuas, de modo que pudéssemos testar uma variedade de técnicas estatísticas básicas. Uma vez que quase todos os conjuntos de dados reais têm pelo menos alguns pontos de dados em falta e uma vez que a capacidade de tratar correctamente os dados em falta é uma das características que consideramos um pacote de análise estatística, introduzimos duas células vazias nos dados: Cada linha da planilha representa um assunto. O primeiro sujeito recebeu o Tratamento 1 e teve o Resultado 1. X e Y são os valores de duas medidas em cada sujeito. Não foi possível obter uma medição para Y no segundo assunto ou em X para o último assunto, portanto essas células estão em branco. Os assuntos são inseridos na ordem em que os dados se tornaram disponíveis, de modo que os dados não são ordenados de nenhuma maneira específica. Usamos esses dados para fazer algumas análises simples e comparamos os resultados com um pacote estatístico padrão. A comparação considerou a precisão dos resultados, bem como a facilidade com que a interface poderia ser usada para conjuntos de dados maiores - ou seja, mais colunas. Utilizamos o SPSS como padrão, embora qualquer um dos pacotes estatísticos que a ILO suporta faria igualmente bem para esse fim. Neste artigo, quando dizemos pacote estatístico de cota, queremos dizer SPSS, SAS, STATA, SYSTAT ou Minitab. A maioria dos procedimentos estatísticos Excels fazem parte do pacote de ferramentas Análise de dados, que está no menu Ferramentas. Inclui uma variedade de escolhas, incluindo estatísticas descritivas simples, testes t, correlações, análise de variância de 1 ou 2 vias, regressão, etc. Se você não tiver um item Análise de Dados no menu Ferramentas, Ferramenta de análise. Procure na Ajuda para quotData Analysis Toolsquot para obter instruções sobre como carregar o ToolPak. Dois outros recursos do Excel são úteis para determinadas análises, mas o pacote de ferramentas Análise de Dados é o único que fornece testes razoavelmente completos de significância estatística. A tabela dinâmica no menu Dados pode ser usada para gerar tabelas de resumo de médias, desvios padrão, contagens, etc. Além disso, você pode usar funções para gerar algumas medidas estatísticas, como um coeficiente de correlação. Funções geram um único número, então usando funções você provavelmente terá que combinar pedaços e bits para obter o que você quer. Mesmo assim, você pode não ser capaz de gerar todas as peças que você precisa para uma análise completa. Salvo indicação em contrário, todos os testes estatísticos usando o Excel foram feitos com o Data Analysis ToolPak. A fim de verificar uma variedade de testes estatísticos, escolhemos as seguintes tarefas: obter médias e desvios padrão de X e Y para todo o grupo e para cada grupo de tratamento. Obter a correlação entre X e Y. Faça um teste t de duas amostras para testar se os dois grupos de tratamento diferem em X e Y. Faça um teste t pareado para testar se X e Y são estatisticamente diferentes uns dos outros. Compare o número de indivíduos com cada resultado pelo grupo de tratamento, usando um teste qui-quadrado. Todas essas tarefas são rotina para um conjunto de dados desta natureza, e todos eles podem ser facilmente feito usando qualquer um dos pacotes listados aobve listados. Questões gerais Habilitar o ToolPak de análise O ToolPak de análise de dados não é instalado com a instalação padrão do Excel. Procure no menu Ferramentas. Se você não tiver um item Análise de Dados, precisará instalar as ferramentas de Análise de Dados. Pesquisar Ajuda para quotData Analysis Toolsquot para obter instruções. Valores ausentes Uma célula em branco é a única maneira para o Excel lidar com dados ausentes. Se você tiver quaisquer outros códigos de valores em falta, será necessário alterá-los para espaços em branco. Arranjo de dados Diferentes análises exigem que os dados sejam organizados de várias maneiras. Se você planejar uma variedade de testes diferentes, pode não haver um único arranjo que funcione. Você provavelmente precisará reorganizar os dados várias maneiras de obter tudo o que você precisa. Caixas de diálogo Escolha ToolsData Analysis e selecione o tipo de análise que você deseja fazer. A caixa de diálogo típica terá os seguintes itens: Intervalo de entrada: digite as células de canto superior esquerdo e inferior direito. por exemplo. A1: B100. Você só pode escolher linhas e colunas adjacentes. A menos que haja uma caixa de seleção para agrupar dados por linhas ou colunas (e geralmente não existe), todos os dados são considerados como um glop. Etiquetas - Às vezes, há uma caixa que você pode marcar para indicar que a primeira linha da folha contém rótulos. Se você tiver rótulos na primeira linha, marque esta caixa e sua saída PODE ser rotulada com o rótulo. Então, novamente, não pode. Local de saída - Nova folha é o padrão. Ou, digite o endereço de célula do canto superior esquerdo de onde você deseja colocar a saída na folha atual. Nova planilha é outra opção, que eu não tentei. As ramificações desta escolha são discutidas abaixo. Outros itens, dependendo da análise. Local de saída A saída de cada análise pode ir para uma nova folha dentro do arquivo do Excel atual (esse é o padrão) ou pode colocá-lo dentro da folha atual especificando a célula de canto superior esquerdo onde deseja colocá-la. De qualquer maneira é um pouco de um incômodo. Se cada saída está em uma nova planilha, você acaba com lotes de folhas, cada uma com um pequeno pedaço de saída. Se você colocá-los na folha atual, você precisa colocá-los adequadamente deixar espaço para adicionar comentários e etiquetas alterações que você precisa fazer para formatar uma saída corretamente pode afetar adversamente outra saída. Exemplo: A saída de Descritivos tem uma coluna de rótulos como Desvio padrão, Erro padrão, etc. Você vai querer fazer essa coluna ampla para poder ler os rótulos. Mas se uma saída de frequência simples estiver bem abaixo, então a coluna que exibe os valores que estão sendo contados, que pode conter números inteiros pequenos, também será ampla. Resultados das análises Estatísticas descritivas A maneira mais rápida de obter médias e desvios padrão para um grupo inteiro é usar Descritivos nas ferramentas Análise de Dados. Você pode escolher várias colunas adjacentes para o intervalo de entrada (neste caso as colunas X e Y), e cada coluna é analisada separadamente. Os rótulos na primeira linha são usados ​​para rotular a saída e as células vazias são ignoradas. Se você tiver mais colunas não adjacentes que você precisa analisar, você terá que repetir o processo para cada grupo de colunas contíguas. O procedimento é direto, pode gerenciar muitas colunas razoavelmente eficiente e células vazias são tratadas adequadamente. Para obter os meios e desvios padrão de X e Y para cada grupo de tratamento, é necessário usar tabelas dinâmicas (a menos que você queira reorganizar a folha de dados para separar os dois grupos). Depois de selecionar o intervalo de dados (contíguo), na opção Layout de assistentes de tabela dinâmica, arraste a área de tratamento para a linha de variável e X para a área de dados. Clique duas vezes em ldquoCount de Xrdquo na área Dados e mude para Média. Arraste X para a caixa Dados novamente, e este tempo mude Contagem para StdDev. Finalmente, arraste X em mais uma vez, deixando-o como Contagem de X. Isso nos dará a Média, o desvio padrão eo número de observações em cada grupo de tratamento para X. Faça o mesmo para Y, então obteremos a média, padrão Desvio e número de observações para Y também. Isso colocará um total de seis itens na caixa Dados (três para X e três para Y). Como você pode ver, se você quiser obter uma variedade de estatísticas descritivas para várias variáveis, o processo ficará tedioso. Um pacote estatístico permite que você escolha quantas variáveis ​​desejar para estatísticas descritivas, sejam ou não contíguas. Você pode obter as estatísticas descritivas para todos os assuntos em conjunto, ou discriminados por uma variável categórica, como tratamento. Você pode selecionar as estatísticas que deseja ver uma vez e ela se aplicará a todas as variáveis ​​escolhidas. Correlações Usando as ferramentas de análise de dados, a caixa de diálogo para correlações é muito parecida com a de descritores - você pode escolher várias colunas contíguas e obter uma matriz de saída de todos os pares de correlações. As células vazias são ignoradas adequadamente. A saída NÃO inclui o número de pares de pontos de dados usados ​​para calcular cada correlação (que pode variar, dependendo de onde você tem dados ausentes) e não indica se alguma das correlações são estatisticamente significativas. Se desejar correlações em colunas não contíguas, você teria que incluir as colunas intermediárias ou copiar as colunas desejadas para um local contíguo. Um pacote estatístico permite que você escolha colunas não-contíguas para suas correlações. A saída indicaria quantos pares de pontos de dados foram usados ​​para calcular cada correlação e quais correlações são estatisticamente significativas. Teste T de duas amostras Este teste pode ser usado para verificar se os dois grupos de tratamento diferem nos valores de X ou Y. Para fazer o teste, você precisa inserir um intervalo de células para cada grupo. Uma vez que os dados não foram inseridos pelo grupo de tratamento, primeiro precisamos ordenar as linhas por tratamento. Certifique-se de tomar todas as outras colunas juntamente com o tratamento, de modo que os dados para cada assunto permanece intacto. Após os dados serem classificados, você pode inserir o intervalo de células que contém as medidas X para cada tratamento. Não inclua a linha com os rótulos, porque o segundo grupo não tem uma linha de etiqueta. Portanto, sua saída não será rotulada para indicar que esta saída é para X. Se você deseja que a saída rotulada, você tem que copiar as células correspondentes ao segundo grupo para uma coluna separada e digite uma linha com um rótulo para o segundo grupo . Se você também quiser fazer o teste t para as medições Y, você precisará repetir o processo. As células vazias são ignoradas e, além dos problemas com a rotulagem da saída, os resultados estão corretos. Um pacote estatístico faria essa tarefa sem qualquer necessidade de classificar os dados ou copiá-los para outra coluna ea saída seria sempre devidamente rotulada na medida em que você fornece rótulos para suas variáveis ​​e grupos de tratamento. Ele também permite que você escolha mais de uma variável de cada vez para o teste t (por exemplo, X e Y). Teste t emparelhado O teste t emparelhado é um método para testar se a diferença entre duas medições sobre o mesmo sujeito é significativamente diferente de 0. Neste exemplo, queremos testar a diferença entre X e Y medidos no mesmo assunto. A característica importante deste teste é que ele compara as medições dentro de cada sujeito. Se você digitalizar as colunas X e Y separadamente, elas não parecem obviamente diferentes. Mas se você olhar para cada par X-Y, você notará que em todos os casos, X é maior que Y. O teste t pareado deve ser sensível a essa diferença. Nos dois casos em que X ou Y está ausente, não é possível comparar as duas medidas em um assunto. Assim, apenas 8 linhas são utilizáveis ​​para o teste t emparelhado. Quando você executa o teste t pareado nestes dados, você obtém uma estatística t de 0,09, com uma probabilidade de 2 caudas de 0,93. O teste não encontra nenhuma diferença significativa entre X e Y. Olhando para a saída com mais cuidado, notamos que ele diz que há 9 observações. Como observado acima, deve haver apenas 8. Parece que o Excel não conseguiu excluir as observações que não tiveram tanto X e Y medições. Para obter os resultados corretos, copie X e Y para duas novas colunas e remova os dados nas células que não têm valor para a outra medida. Agora volte a executar o teste t emparelhado. Desta vez, a estatística t é 6,14817 com uma probabilidade de 2 caudas de 0,000468. A conclusão é completamente diferente Claro, este é um exemplo extremo. Mas o ponto é que o Excel não calcula o t-teste emparelhado corretamente quando algumas observações têm uma das medidas, mas não a outra. Embora seja possível obter o resultado correto, você não teria motivo para suspeitar dos resultados obtidos a menos que esteja suficientemente alerta para perceber que o número de observações está errado. Não existe nada na ajuda online que o avise sobre este problema. Curiosamente, há também uma função TTEST, que dá os resultados corretos para este exemplo. Aparentemente, as funções e as ferramentas de análise de dados não são consistentes na forma como lidam com células ausentes. No entanto, não posso recomendar o uso de funções de preferência às ferramentas de Análise de Dados, porque o resultado de usar uma função é um único número - neste caso, a probabilidade de 2 caudas da estatística t. A função não lhe dá a t-estatística em si, os graus de liberdade, ou qualquer número de outros itens que você gostaria de ver se você estava fazendo um teste estatístico. Um pacote estatístico excluirá corretamente os casos com uma das medidas ausentes e fornecerá todas as estatísticas de suporte necessárias para interpretar a saída. Crosstabulation e Qui-Quadrado Teste de Independência Nossa tarefa final é contar os dois resultados em cada grupo de tratamento e usar um qui-quadrado teste de independência para testar uma relação entre tratamento e desfecho. A fim de contar os resultados por grupo de tratamento, você precisa usar tabelas dinâmicas. Na opção Layout de Assistentes de Tabela Dinâmica, arraste Tratamento para Fila, Resultado para a Coluna e também para Dados. A área de dados deve dizer quotCount de Outcomequot ndash se não, clique duas vezes nele e selecione quotCountquot. Se desejar porcentagens, clique duas vezes em quotCount de Outcomequot e clique em Opções na caixa ldquoShow Data Asrdquo que aparece, selecione quot de rowquot. Se você quiser ambas as contagens e porcentagens, você pode arrastar a mesma variável para a área de dados duas vezes, e usá-lo uma vez para contagens e uma vez para porcentagens. Obter o teste qui-quadrado não é tão simples, no entanto. Ele só está disponível como uma função ea entrada necessária para a função é a contagem observada em cada combinação de tratamento e resultado (que você tem em sua tabela dinâmica) e as contagens esperadas em cada combinação. Contas esperadas Como são obtidas Se você tiver conhecimento estatístico suficiente para saber como calcular as contagens esperadas e pode fazer cálculos do Excel usando endereços de célula relativa e absoluta, você deve ser capaz de navegar por isso. Se não, você está fora de sorte. Assumindo que você superou o problema das contagens esperadas, você pode usar a função de Chitest para obter a probabilidade de observar um valor de qui-quadrado maior do que aquele para esta tabela. Novamente, uma vez que estamos usando funções, você não recebe muitas outras peças necessárias do cálculo, notavelmente o valor da estatística chi-quadrado ou seus graus de liberdade. Nenhum pacote estatístico exigiria que você fornecesse os valores esperados antes de calcular um teste qui-quadrado de independência. Além disso, os resultados sempre incluem a estatística chi-quadrado e seus graus de liberdade, bem como sua probabilidade. Muitas vezes você vai ter algumas estatísticas adicionais também. Análises Adicionais As análises restantes não foram feitas neste conjunto de dados, mas alguns comentários sobre eles são incluídos para completar. Frequências simples Você pode usar tabelas dinâmicas para obter freqüências simples. (Consulte Crosstabulations para obter mais informações sobre como obter tabelas dinâmicas.) Usando tabelas dinâmicas, cada coluna é considerada uma variável separada e rótulos na linha 1 aparecerão na saída. Você só pode fazer uma variável de cada vez. Outra possibilidade é usar a função Frequências. A principal vantagem deste método é que uma vez que você definiu a função de freqüências para uma coluna, você pode usar CopyPaste para obtê-lo para outras colunas. Primeiro, você precisará digitar uma coluna com os valores que deseja contar (bins). Se você pretende fazer as freqüências para muitas colunas, certifique-se de inserir valores para a coluna com a maioria das categorias. por exemplo. Se 3 colunas tiverem valores de 1 ou 2 e o quarto tiver valores de 1,2,3,4, você precisará digitar os valores de bin como 1,2,3,4. Agora selecione suficientes células vazias em uma coluna para armazenar os resultados - 4 neste exemplo, mesmo que a coluna atual tenha apenas 2 valores. Em seguida, escolha InsertFunctionStatisticalFrequencies no menu. Preencha o intervalo de entrada para a primeira coluna que você deseja contar usando endereços relativos (por exemplo A1: A100). Preencha o Bin Range usando os endereços absolutos dos locais onde você digitou os valores a serem contados (por exemplo, M1: M4). Clique em Concluir. Observe a caixa acima dos cabeçalhos de coluna da folha, onde a fórmula é exibida. Começa com quot FREQUENCIES Coloque o cursor à esquerda do sinal na fórmula e pressione Ctrl-Shift-Enter. A frequência de contagem agora aparece nas células que você selecionou. Para obter as contagens de freqüência de outras colunas, selecione As células com as freqüências neles e escolha EditCopy no menu. Se a próxima coluna que você deseja contar é uma coluna à direita do anterior, selecione a célula à direita da primeira célula de freqüência e escolha EditPaste Ctrl-V) Continue movendo para a direita e colando para cada coluna que você quer contar Cada vez que você move uma coluna para a direita das células de freqüência original, a coluna a ser contada é deslocada para a direita da primeira coluna que você contou. Se você quiser porcentagens também, yoursquoll tem que usar a função Sum para calcular a soma das freqüências e definir a fórmula para obter a porcentagem de uma célula. Consulte a célula para armazenar o primeiro por cento e digite a fórmula na fórmula Caixa na parte superior da folha - eg N1100N 5 - onde N1 é a célula com a freqüência para a primeira categoria, e N5 é a célula com a soma das freqüências. Use CopyPaste para obter a fórmula para as células restantes da primeira coluna. Depois de ter os percentuais para uma coluna, você pode Copiar-los para as outras colunas. Yoursquoll precisa ter cuidado com o uso de endereços relativos e absolutos No exemplo acima, usamos N5 para o denominador, então quando copiar a fórmula para baixo para a próxima freqüência na mesma coluna, ele ainda vai olhar para a soma na linha 5, mas quando copiar a fórmula para a direita para outra coluna, ele irá mudar para as freqüências na próxima coluna. Finalmente, você pode usar Histograma no menu Análise de dados. Você só pode fazer uma variável de cada vez. Como com a função Freqüências, você deve inserir uma coluna com limites quotbinquot. Para contar o número de ocorrências de 1 e 2, você precisará inserir 0,1,2 em três células adjacentes e atribuir o intervalo dessas três células como as caixas na caixa de diálogo. A saída não é rotulada com quaisquer etiquetas que você pode ter na linha 1, nem mesmo com a letra da coluna. Se você fizer freqüências em muitas variáveis, você terá dificuldade em saber qual freqüência pertence a qual coluna de dados. Regressão linear Uma vez que a regressão é uma das análises estatísticas mais utilizadas, nós tentamos fazê-lo, mesmo que não fizemos uma análise de regressão para este exemplo. O procedimento de Regressão nas ferramentas Análise de Dados permite escolher uma coluna como a variável dependente e um conjunto de colunas contíguas para os independentes. No entanto, ele não tolera quaisquer células vazias em qualquer lugar nos intervalos de entrada, e você está limitado a 16 variáveis ​​independentes. Portanto, se você tiver quaisquer células vazias, você precisará copiar todas as colunas envolvidas na regressão para novas colunas e excluir quaisquer linhas que contenham quaisquer células vazias. Modelos grandes, com mais de 16 preditores, não podem ser feitos de forma alguma. Análise de Variância Em geral, as características de ANOVA Excels estão limitadas a alguns casos especiais raramente encontrados fora dos manuais e requerem muitos reajustes de dados. ANOVA unidirecional Os dados devem ser dispostos em colunas (ou linhas) separadas e adjacentes para cada grupo. Claramente, isso não é propício para fazer 1-maneiras em mais de um agrupamento. Se você tiver rótulos na linha 1, a saída usará as etiquetas. ANOVA de Dois Factores Sem Replicação Isto só faz o caso com uma observação por célula (isto é, sem termo de erro dentro da célula). O intervalo de entrada é um arranjo retangular de células, com linhas representando níveis de um fator, colunas os níveis do outro fator e o conteúdo da célula o único valor nessa célula. ANOVA de dois fatores com réplicas Isso faz uma ANOVA de duas vias com tamanhos de célula iguais. A entrada deve ser uma região retangular com colunas que representem os níveis de um fator e as linhas que representem repetições dentro dos níveis do outro fator. A faixa de entrada deve também incluir uma linha adicional na parte superior e coluna à esquerda, com rótulos indicando os fatores. No entanto, esses rótulos não são usados ​​para rotular a tabela ANOVA resultante. Clique em Ajuda na caixa de diálogo ANOVA para obter uma imagem do que o intervalo de entrada deve ser semelhante. Solicitando muitas análises Se você tivesse uma variedade de diferentes procedimentos estatísticos que você queria executar em seus dados, você quase certamente se veria fazendo um monte de classificação, reorganização, cópia e colagem de seus dados. Isso ocorre porque cada procedimento requer que os dados sejam dispostos de uma maneira particular, muitas vezes diferente da maneira como outro procedimento deseja que os dados sejam dispostos. Em nosso pequeno teste, tivemos que classificar as linhas para fazer o teste t, e copiar algumas células para obter rótulos para a saída. Tínhamos que limpar o conteúdo de algumas células para obter o teste t pareado correto, mas não queríamos que essas células fossem limpas para algum outro teste. E nós estávamos apenas fazendo cinco tarefas. Ele não fica melhor quando você tenta fazer mais. Não há um arranjo único dos dados que permitiria que você fizesse muitas análises diferentes sem fazer muitas cópias diferentes dos dados. A necessidade de manipular os dados de muitas maneiras aumenta a chance de introduzir erros. Usando um programa estatístico, os dados seriam normalmente organizados com as linhas que representam os sujeitos e as colunas que representam as variáveis ​​(como estão em nossos dados de amostra). Com este arranjo você pode fazer qualquer uma das análises discutidas aqui, e muitos outros também, sem ter que classificar ou reorganizar seus dados de forma alguma. Somente análises muito mais complexas, além das capacidades do Excel e do escopo deste artigo exigiria o rearranjo dos dados. Trabalhando com muitas colunas E se seus dados não tivessem 4, mas 40 colunas, com uma mistura de medidas categóricas e contínuas Como facilmente os procedimentos acima se dimensionam para um problema maior Na melhor das hipóteses, alguns dos procedimentos estatísticos podem aceitar várias colunas contíguas para entrada , E interpretar cada coluna como uma medida diferente. Os procedimentos descritivos e de correlações são deste tipo, portanto, você pode solicitar estatísticas descritivas ou correlações para um grande número de variáveis ​​contínuas, desde que sejam inseridas em colunas adjacentes. Se eles não são adjacentes, você precisa reorganizar colunas ou usar copiar e colar para torná-los adjacentes. Muitos procedimentos, no entanto, só podem ser aplicados a uma coluna de cada vez. T-testes (independentes ou emparelhados), contagens de freqüência simples, o qui-quadrado teste de independência, e muitos outros procedimentos estão nesta classe. Isso se tornaria uma séria desvantagem se você tivesse mais do que um punhado de colunas, mesmo se você usar cortar e colar ou macros para reduzir o trabalho. Além de ter que repetir o pedido muitas vezes, você tem que decidir onde armazenar os resultados de cada um, e verifique se ele está corretamente rotulado para que você possa facilmente localizar e identificar cada saída. Finalmente, o Excel não fornece um registro ou outro registro para rastrear o que você fez. Isso pode ser um inconveniente grave se você quiser ser capaz de repetir a mesma análise (ou similar) no futuro, ou mesmo se você simplesmente esqueceu o que já fez. Usando um pacote estatístico, você pode solicitar um teste para tantas variáveis ​​como você precisa de uma vez. Cada um será devidamente rotulado e organizado na saída, por isso não há confusão quanto ao que é o quê. Você também pode esperar para obter um log, e muitas vezes um conjunto de comandos também, que pode ser usado para documentar o seu trabalho ou para repetir uma análise sem ter que passar por todas as etapas novamente. Embora o Excel seja uma planilha fina, não é um pacote de análise de dados estatísticos. Com toda a justiça, nunca foi pretendido ser um. Tenha em mente que o Data Analysis ToolPak é um quotadd-inquot - um recurso extra que permite que você faça alguns cálculos rápidos. Portanto, não deve ser surpreendente que isso é apenas o que é bom para - alguns cálculos rápidos. Se você tentar usá-lo para análises mais extensas, você encontrará dificuldades devido a uma ou todas as seguintes limitações: Problemas potenciais com análises envolvendo dados ausentes. Estes podem ser insidiosos, em que o usuário incautável é improvável que perceber que algo está errado. Falta de flexibilidade nas análises que podem ser feitas devido às suas expectativas quanto ao arranjo dos dados. Isso resulta na necessidade de cortar o sabor e reorganizar a folha de dados de várias maneiras, aumentando a probabilidade de erros. Saída espalhada em muitas planilhas diferentes, ou em toda uma planilha, que você deve assumir a responsabilidade de organizar de forma sensata. A saída pode estar incompleta ou não estar devidamente rotulada, aumentando a possibilidade de erro de identificação da saída. Necessidade de repetir solicitações para algumas análises várias vezes para executá-lo para várias variáveis, ou para solicitar várias opções. Necessidade de fazer algumas coisas definindo suas próprias funções formulas, com o seu risco de erros. Nenhum registro do que você fez para gerar seus resultados, tornando difícil documentar sua análise, ou repeti-la em um momento posterior, se isso for necessário. Se você tiver mais do que cerca de 10 ou 12 colunas, e ou quiser fazer qualquer coisa além de estatísticas descritivas e talvez correlações, você deve usar um pacote estatístico. Há vários disponíveis por licença de site através da OIT, ou você pode usá-los em qualquer laboratório da OIT. Se você tiver o Excel em seu próprio PC e não quiser pagar por um programa estatístico, use o Excel para digitar os dados (com linhas representando os assuntos e colunas para as variáveis). Todos os pacotes estatísticos mencionados podem ler arquivos do Excel, assim você pode fazer a entrada de dados (demorada) em casa, e ir para os laboratórios para fazer a análise. Uma discussão muito mais extensa sobre as armadilhas do uso do Excel, com muitos links adicionais, está disponível em burns-stat Clique em Tutoriais e, em seguida, em Spreadsheet Addiction. Para obter assistência ou obter mais informações sobre software estatístico, entre em contato com o Centro de Consultoria de Bioestatística. Telefone 545-2949From 8:00 PM CST sexta-feira, fevereiro 24 - 6:00 PM CST Sábado, fevereiro 25, ni estará passando por atualizações de sistema que podem resultar em interrupção temporária de serviço. Agradecemos a sua paciência à medida que melhoramos a nossa experiência online. Mais do que o Microsoft Excel para medição Análise de dados e relatórios Data de publicação: 18, 2014 44 4,39 5 Imprimir Devido principalmente à sua ampla disponibilidade, o Microsoft Excel é muitas vezes a escolha de facto de engenheiros e cientistas que necessitam de software para análise de dados de medição e manipulação. O Microsoft Excel se presta bem a aplicações de teste e medição extremamente simples e aos usos financeiros para os quais foi projetado, porém, numa época em que as empresas são forçadas a fazer mais com menos, é imperativo escolher as ferramentas adequadas para maximizar a eficiência . Só porque o Microsoft Excel já está instalado no seu computador não o torna a ferramenta certa para cada trabalho. O software DIAdem da National Instruments, criado especificamente para a gestão, inspeção, análise e geração de relatórios de dados científicos e de engenharia adquiridos ou simulados, oferece ganhos de eficiência e escalabilidade com recursos que superam as limitações do Excel na maioria dos aplicativos de pós-processamento de dados. 1. Diferenças em Blocos de Construção Fundamentais: Células versus Canais O Microsoft Excel usa a célula como seu bloco de construção fundamental.160 Células formam linhas e colunas para compor uma planilha, uma arquitetura ideal para orçamentos e balanços.160 Simples , Aplicações de aquisição de dados de ponto único, por exemplo, aquelas que coletam um único ponto de dados por hora ao longo de um dia são freqüentemente mapeadas para esta arquitetura porque cada ponto de dados individual tem mais importância quando menos pontos de dados são coletados. O ponto existe como uma célula em uma planilha e deve ser manipulado usando o paradigma Excels baseado em células. A maioria das aplicações de aquisição de dados, no entanto, não é trivial.160 As aplicações que coletam dezenas de canais de dados a taxas de mega-amostra por segundo (MSs) são comuns.160 Nessas aplicações, a manipulação e interação de dados é feita em um sinal ou canal como Ao manipular canais no Excel como colunas de células individuais, a unidade de um sinal é perdida.160 Embora colunas completas do Excel possam ser manipuladas de cada vez, isso é mais complicado com colunas mais longas.160 Além disso, as colunas contêm muitas vezes descritivas Informações como um nome ou unidade além dos dados numéricos brutos.160 Nesse caso, um subconjunto da coluna deve ser selecionado (por exemplo, o intervalo A2: A99), introduzindo sobrecarga e o potencial de imprecisão ou erros. Na Figura 1. O Excel é usado para executar uma tarefa de engenharia simples mas comum: calcular a média de cinco canais de temperatura armazenados em colunas para criar um canal médio resultante.160 O cálculo da média deve ser feito primeiro com o bloco de construção de uma célula e então copiado (ou preenchido) para all cells in the resultant column.160 Using DIAdem, which uses the channel as its fundamental building block, averaging channels is as simple as dragging-and-dropping input channels to the Average Channels function, as shown in Figure 2 .160 Individual data points can still be manipulated in DIAdem when necessary. Figure 1 . Microsoft Excel uses the cell as its fundamental building block.160 Even simple data analysis must be applied to a cell and then repeated for all cells in a column (channel). Figure 2 . NI DIAdem operates with the building block of a160channel.160 Averaging is as simple as dragging-and-dropping entire data channels instead of unnecessarily manipulating individual data points. 2. Hundreds of Engineering and Scientific Analysis Calculations While the number of available formulas for finance-oriented calculations in Excel is extensive, you must configure an optional add-in called the Analysis Toolpak for access to a few engineering and statistical calculations.160 The Analysis Toolpak functions are extremely limited, as shown in Figure 3 with the common engineering calculation Fast Fourier Transform (FFT).160 In general, Excels analysis capabilities often do not meet the requirements of scientific or engineering applications.160 For further extensibility, Excel provides a robust Visual Basic for Applications (VBA) engine and an excellent VBA editing environment that enable you to write your own engineering calculations from scratch when Excels built in functions are insufficient for your application. Figure 3 . Microsoft Excel provides an exhaustive set of finance-based calculations and allows engineers to write their own code to meet their application needs. In DIAdem, hundreds of engineering and scientific analysis calculations from simple addition to complex matrix manipulation and order analysis have been included in the environment.160 The analysis functions in DIAdem are configuration-based no programming is required to run even complex analysis such as Digital Filtering, as shown in Figure 4 .160 Additionally, DIAdem analysis functions include full previews of analysis results so that you can avoid erroneous calculations by ensuring that youre using the correct parameters prior to running each calculation. Figure 4 . DIAdem includes hundreds of analysis functions specific to scientists and engineers.160 Each calculation is configuration-based and provides a preview of resultant channels so that you can interact with parameters to ensure accuracy and reduce errors. DIAdem also includes a framework for creating your own domain-specific calculations called the Calculation Manager, and it includes an integrated Visual Basic scripting interface for sequencing built-in DIAdem calculations or defining your own custom calculations. 3. Loading and Manipulating Large Volumes of Data Data streaming speeds of common applications reach or exceed MSs rates.160 In an application that collects one single channel of data at 1 MSs, a total of 1,000,000 data points will be collected in a one second acquisition. In a matter of minutes, billions of data points can be saved to gigabytes of hard drive space. When Microsoft Excel attempts to load a data file containing a large volume of data, it attempts to load every single data point into memory.160 With the release of the 64-bit version of Microsoft Excel 2010, this is less of a limitation, as the application has a larger addressable memory space however, loading the entirety of a large data set into Excel can often take many minutes due to the sheer volume of data that needs to be loaded.160 Furthermore, Excel stores not just numerical values in each cell but also numeric formatting, cell formatting, formulas, spreadsheet links, Internet hyperlinks, and comments.160 This cell-centric flexibility is ideal for business spreadsheets where cell-level visibility is key, but it adds160unnecessary memory overhead for data sets with millions of values. 160To avoid potential memory problems, Excel imposes a limit on the maximum number of 160rows and columns.160 The introduction of Excel 2007 increased the total number of rows per worksheet from 65,536 to just over 1,000,000 (2 20. to be precise) and the total number of columns from 256 to 16,384 (2 14 ).160 Using Figures 5 and 6 . contrast Excels row and column limitation with DIAdems ability to manipulate 500,000,000 rows (points) as only a fraction of its limitation.160 Figure 5 . Excel can only load just over 1 million rows of data for any given column.160 This is a limitation for scientists and engineers. Figure 6 . DIAdem can easily handle extremely large data sets.160 This image shows 160an example of 500,000,000 (one-half Billion) data points in a channel - 500 times the maximum number of rows allowed by Excel. As shown in Figure 5 . an acquisition rate of 1 MSs using one single channel would exceed the number of data points that Excel could load in just over one second of acquisition.160 Many engineers and scientists feel forced to allow the limitations of their data post-processing software to dictate the terms of their acquisition and either reduce acquisition rates or segment acquisitions across numerous data files, introducing a nightmare for data management and organization. DIAdem was designed to manipulate measurement data in both small and large volumes, and can process up to 2,000,000,000 data points (2 31 ) per channel across 65,536 (2 16 ) total data channels.160 Additionally, DIAdem includes160 selective loading, data reduction and register loading features specifically designed for working with extremely large data sets.160 DIAdem can selectively load a subset of the data channels contained in a data file, whereas Excel always imports all of the columns from a data file.160 If you only need to load 1 channel from a very large data file with 10 channels in it, loading only the 10 of the data values that you actually need is much faster and more efficient than Excels method of loading 100 of the data when 90 is overhead.160 When files are loaded with data reduction, DIAdem loads data from a selected row range andor condenses every N rows into one representative value, whereas Excel always loads all the data rows.160 When files are register loaded, DI Adem uses the existing data file on disk as in-place virtual memoryDIAdem does not load all the values from the data file at once but instead registers how to access blocks of data values on-demand.160 This makes register loaded channels 160read-only, but it enables very quick graphing and inspection of extremely large data sets, as shown in Figure 6 . View a user solution on how DIAdem is processing massive amounts of data to help predict and monitor earthquake activity. 4. Flexibility in File Storage Format Applications collecting and saving data at high streaming rates must write data to disk using a streaming-capable file format.160 Binary file formats are most often utilized because they do not include the extraneous overhead required to make a file human-readable the way ASCII files do.160 A comparison of common file formats, 160including the open binary format standard to National Instruments software called Technical Data Management Streaming (TDMS), is displayed in Table 1 . 160160160160160 Table 1 . There are many file format options available, but binary file formats such as TDMS are the only formats that are high-speed streaming capable. May require a toolkit or add-on module. DIAdem is flexible enough to read any custom file format including customized binary file formats due to modular pieces of code called DataPlugins that know how to parse and interpret 160the contents of a data file.160 DataPlugins also reformat the parsed data from the particular data file into a common data structure inside DIAdem, which makes it easy to compare data loaded from different file formats.160 National Instruments has published free downloadable DataPlugins for hundreds of the most commonly used data file formats, and there are published APIs for LabVIEW and VBScript for you to create DataPlugins for your own legacy data files.160 This enables DIAdem to be modular and scalable enough to handle any current, legacy, or future data file format choices. By contrast, while Excel can read ASCII files, it is usually completely unable to load data from binary files.160 Even when Excel successfully loads an ASCII file, it has limited ability to correctly interpret the property and channel structure of the data file. 160Too often, time-consuming reformatting of the imported ASCII data is necessary before it can be used at all.160 One exception to this problem is the 160TDMS file format from 160National Instruments, which Excel loads with correct formatting of the structure, properties, and data from the TDMS file, using160 the free TDM Excel Add-In . Visit the TDMS homepage to learn more about the TDMS file format for storing measurement data to disk. 5. Built-In Tools for Data Management and Trending Over time, it is common to store and attempt to organize hundreds or thousands of data files on disk.160 These files are often stored in different ways using different formats, and may even be stored in varying locations on a local machine or across a network. If you want to trend data across multiple data files using Microsoft Excel, you have to open each individual data file, copy the pertinent columns and paste them into a master (aggregated) file, and move onto the next data file.160 Accurately trending similar data channels across hundreds of unique data files could take days or weeks. DIAdem can accomplish this same task in seconds.160 Using DataPlugins, DIAdem can load these different file formats to a common imported structure for uniform analysis and reporting. Furthermore, DIAdem installs a technology called My DataFinder that helps you quickly locate and isolate the exact data sets you are looking for, even if they are located across different files as shown in Figure 7. 160 My DataFinder automatically creates an index of the descriptive information contained within data files which becomes searchable within DIAdem.160 Using DIAdem and DataFinder technology, you can quickly find all data files that were written by a particular operator, locate all failed tests, or even identify any data channels across all data files that were stored using a specific type of sensor.160 The more information you document in your data files, the more possibilities are available when searching for specific data stored across multiple files in various folders andor file formats. Figure 7 . In this query, the DataFinder has located data channels across all data files that were collected using a J-Type thermocouple and stored to disk by Jennifer, the operator. View a webcast demonstrating DIAdem and the NI DataFinder for data management, analysis, and reporting. 6. Data Inspection and Synchronization Microsoft Excel allows users to create basic charts and graphs, but static graphs do not allow you to fully interact with and inspect data that has been measured over time.160 For example, it is impossible to visualize correlated measurement data and GPS160data using built-in tools in Excel. 160In order to completely characterize all aspects of time-based measurements, DIAdem includes a powerful visualization tool that features fully-synchronizable display areas ideal for playing back measurements coordinated with videos, 3D models, axis systems, GPS map displays, contours, sound playback, and more.160 This enables you to replay a measurement synchronized with other information to completely understand its context.160 DIAdems visualization tool also enables you to easily zoom into a specific region of a graph, copy or delete or interpolate data ranges, and examine the exact values of specific points graphically.160 Using this dynamic tool, it is easy to identify regions of interest or locate outliers within larger data sets. Figure 8 . Using DIAdem, you can fully synchronize the playback of measurement data, sound data, GPS coordinates, video, and more . 7. What-You-See-Is-What-You-Get (WYSIWYG) Reporting Templates DIAdem features a robust reporting engine that leverages reusable templates because many engineers generate the same reports repeatedly using different data sets.160 The WYSIWYG report templates in DIAdem store references to data in memory as opposed to storing the actual data values themselves. To create reports of different data sets using the same stored report template, you can simply load the new data into memory and the loaded report template instantly updates 160its display with the newly loaded data values.160 You can then export completed high-resolution, publication-ready reports to the most common reporting formats including PDF, PowerPoint, HTML, image, and more. 160In Excel, the report display is saved along with the data in a common spreadsheet file, which makes it much harder to use a particular report display for multiple data sets. 160 Figure 9 . DIAdem features a WYSIWYG report editor publication-ready exported reports will look identical to their edit-time templates. 8. Interactive Automation Excel provides a powerful environment for the development of macros.160 Using recording mode, it is possible to interactively record macros that 160automate lengthy evaluations or calculations.160 DIAdem similarly160 features 160an integrated VBScript editor, a user dialog editor, and a script recorder to interactively generate scripts that automate lengthy evaluations or calculations. 160160Using scripting, the entire DIAdem environment can be customized and automated so that repetitive data workflow processes that used to take days can be accomplished in a matter of minutes.160 This truly maximizes the efficiency of engineers and scientists and dramatically reduces the time it takes them to turn raw measurement data into usable information. 9. Excel is Free, Yet Too Costly to Use Microsoft Office is used by approximately 80 of enterprises1 .160 Many engineers and scientists view Excel as free software because it is installed on most enterprise computers without question.160 Often, scientists and engineers begin to use Excel for their analysis and reporting needs because it is familiar and available.160 When they encounter Excels limitations as summarized in Table 2 . they either live with often repetitive and time consuming manual processing or spend hours and weeks developing and maintaining custom macro code. If you estimate that personnel costs (including salary, insurance, equipment, etc) for one engineer total 100,000 annually, the cost to purchase one license of DIAdem Advanced and one entire week of training would be recuperated after just 2.8 work weeks of realized productivity gains over Microsoft Excel. As detailed in this document, NI DIAdem quickly pays for itself by overcoming the limitations of Excel and introducing additional efficiency tools for managing, analyzing and160 reporting measurement data.160160 Table 2 . DIAdem will increase your efficiency by overcoming the limitations of Microsoft Excel. 10. Learn More and Move to DIAdem Today Use the resources below to learn more about moving beyond Excel to more powerful tools for measurement data analysis and reporting. View a webcast highlighting DIAdems benefits for data management, analysis, and reporting. Watch six 1-minute videos to learn more about DIAdem. Download DIAdem and explore the environment with a free, 7-day evaluation . Speak to an expert for a free web demo or to have your questions answered immediately. Excel For Statistical Data Analysis This is a webtext companion site of Business Statistics USA Site Para mis visitantes del mundo de habla hispana, este sitio se encuentra disponible en espaol en: Sitio Espejo para Amrica Latina Sitio de los E. E.U. U. Excel is the widely used statistical package, which serves as a tool to understand statistical concepts and computation to check your hand-worked calculation in solving your homework problems. The site provides an introduction to understand the basics of and working with the Excel. Redoing the illustrated numerical examples in this site will help improving your familiarity and as a result increase the effectiveness and efficiency of your process in statistics. To search the site . try E dit F ind in page Ctrl f. Enter a word or phrase in the dialogue box, e. g. quot variancequot or quot meanquot If the first appearance of the wordphrase is not what you are looking for, try F ind Next. Introduction This site provides illustrative experience in the use of Excel for data summary, presentation, and for other basic statistical analysis. I believe the popular use of Excel is on the areas where Excel really can excel. This includes organizing data, i. e. basic data management, tabulation and graphics. For real statistical analysis on must learn using the professional commercial statistical packages such as SAS, and SPSS. Microsoft Excel 2000 (version 9) provides a set of data analysis tools called the Analysis ToolPak which you can use to save steps when you develop complex statistical analyses. You provide the data and parameters for each analysis the tool uses the appropriate statistical macro functions and then displays the results in an output table. Some tools generate charts in addition to output tables. If the Data Analysis command is selectable on the Tools menu, then the Analysis ToolPak is installed on your system. However, if the Data Analysis command is not on the Tools menu, you need to install the Analysis ToolPak by doing the following: Step 1: On the Tools menu, click Add-Ins. If Analysis ToolPak is not listed in the Add-Ins dialog box, click Browse and locate the drive, folder name, and file name for the Analysis ToolPak Add-in Analys32.xll usually located in the Program FilesMicrosoft OfficeOfficeLibraryAnalysis folder. Once you find the file, select it and click OK. Step 2: If you dont find the Analys32.xll file, then you must install it. Insert your Microsoft Office 2000 Disk 1 into the CD ROM drive. Select Run from the Windows Start menu. Browse and select the drive for your CD. Select Setup. exe, click Open, and click OK. Click the Add or Remove Features button. Click the next to Microsoft Excel for Windows. Click the next to Add-ins. Click the down arrow next to Analysis ToolPak. Select Run from My Computer. Select the Update Now button. Excel will now update your system to include Analysis ToolPak. Launch Excel. On the Tools menu, click Add-Ins. - and select the Analysis ToolPak check box. Step 3: The Analysis ToolPak Add-In is now installed and Data Analysis. will now be selectable on the Tools menu. Microsoft Excel is a powerful spreadsheet package available for Microsoft Windows and the Apple Macintosh. Spreadsheet software is used to store information in columns and rows which can then be organized andor processed. Spreadsheets are designed to work well with numbers but often include text. Excel organizes your work into workbooks each workbook can contain many worksheets worksheets are used to list and analyze data. Excel is available on all public-access PCs (i. e. those, e. g. in the Library and PC Labs). It can be opened either by selecting Start - Programs - Microsoft Excel or by clicking on the Excel Short Cut which is either on your desktop, or on any PC, or on the Office Tool bar. Opening a Document: Click on File-Open (CtrlO) to openretrieve an existing workbook change the directory area or drive to look for files in other locations To create a new workbook, click on File-New-Blank Document. Saving and Closing a Document: To save your document with its current filename, location and file format either click on File - Save. If you are saving for the first time, click File-Save choosetype a name for your document then click OK. Also use File-Save if you want to save to a different filenamelocation. When you have finished working on a document you should close it. Go to the File menu and click on Close. If you have made any changes since the file was last saved, you will be asked if you wish to save them. The Excel screen Workbooks and worksheets: When you start Excel, a blank worksheet is displayed which consists of a multiple grid of cells with numbered rows down the page and alphabetically-titled columns across the page. Each cell is referenced by its coordinates (e. g. A3 is used to refer to the cell in column A and row 3 B10:B20 is used to refer to the range of cells in column B and rows 10 through 20). Your work is stored in an Excel file called a workbook. Each workbook may contain several worksheets andor charts - the current worksheet is called the active sheet. To view a different worksheet in a workbook click the appropriate Sheet Tab. You can access and execute commands directly from the main menu or you can point to one of the toolbar buttons (the display box that appears below the button, when you place the cursor over it, indicates the nameaction of the button) and click once. Moving Around the Worksheet: It is important to be able to move around the worksheet effectively because you can only enter or change data at the position of the cursor. You can move the cursor by using the arrow keys or by moving the mouse to the required cell and clicking. Once selected the cell becomes the active cell and is identified by a thick border only one cell can be active at a time. To move from one worksheet to another click the sheet tabs. (If your workbook contains many sheets, right-click the tab scrolling buttons then click the sheet you want.) The name of the active sheet is shown in bold. Moving Between Cells: Here is a keyboard shortcuts to move the active cell: Home - moves to the first column in the current row CtrlHome - moves to the top left corner of the document End then Home - moves to the last cell in the document To move between cells on a worksheet, click any cell or use the arrow keys. To see a different area of the sheet, use the scroll bars and click on the arrows or the area abovebelow the scroll box in either the vertical or horizontal scroll bars. Note that the size of a scroll box indicates the proportional amount of the used area of the sheet that is visible in the window. The position of a scroll box indicates the relative location of the visible area within the worksheet. Entering Data A new worksheet is a grid of rows and columns . The rows are labeled with numbers, and the columns are labeled with letters. Each intersection of a row and a column is a cell . Each cell has an address . which is the column letter and the row number. The arrow on the worksheet to the right points to cell A1, which is currently highlighted . indicating that it is an active cell . A cell must be active to enter information into it. To highlight (select) a cell, click on it. To select more than one cell: Click on a cell (e. g. A1), then hold the shift key while you click on another (e. g. D4) to select all cells between and including A1 and D4. Click on a cell (e. g. A1) and drag the mouse across the desired range, unclicking on another cell (e. g. D4) to select all cells between and including A1 and D4.To select several cells which are not adjacent, press control and click on the cells you want to select. Click a number or letter labeling a row or column to select that entire row or column. One worksheet can have up to 256 columns and 65,536 rows, so itll be a while before you run out of space. Each cell can contain a label . value . logical value . or formula . Labels can contain any combination of letters, numbers, or symbols. Values are numbers. Only values (numbers) can be used in calculations. A value can also be a date or a timeLogical values are true or false. Formulas automatically do calculations on the values in other specified cells and display the result in the cell in which the formula is entered (for example, you can specify that cell D3 is to contain the sum of the numbers in B3 and C3 the number displayed in D3 will then be a funtion of the numbers entered into B3 and C3). To enter information into a cell, select the cell and begin typing. Note that as you type information into the cell, the information you enter also displays in the formula bar. You can also enter information into the formula bar, and the information will appear in the selected cell. When you have finished entering the label or value: Press Enter to move to the next cell below (in this case, A2) Press Tab to move to the next cell to the right (in this case, B1)Click in any cell to select it Entering Labels Unless the information you enter is formatted as a value or a formula, Excel will interpret it as a label, and defaults to align the text on the left side of the cell. If you are creating a long worksheet and you will be repeating the same label information in many different cells, you can use the AutoComplete function. This function will look at other entries in the same column and attempt to match a previous entry with your current entry. For example, if you have already typed Wesleyan in another cell and you type W in a new cell, Excel will automatically enter Wesleyan. If you intended to type Wesleyan into the cell, your task is done, and you can move on to the next cell. If you intended to type something else, e. g. Williams, into the cell, just continue typing to enter the term. To turn on the AutoComplete funtion, click on Tools in the menu bar, then select Options, then select Edit, and click to put a check in the box beside Enable AutoComplete for cell values. Another way to quickly enter repeated labels is to use the Pick List feature. Right click on a cell, then select Pick From List. This will give you a menu of all other entries in cells in that column. Click on an item in the menu to enter it into the currently selected cell. A value is a number, date, or time, plus a few symbols if necessary to further define the numbers 91such as. - ( ) 93. Numbers are assumed to be positive to enter a negative number, use a minus sign - or enclose the number in parentheses (). Dates are stored as MMDDYYYY, but you do not have to enter it precisely in that format. If you enter jan 9 or jan-9, Excel will recognize it at January 9 of the current year, and store it as 192002. Enter the four-digit year for a year other than the current year (e. g. jan 9, 1999). To enter the current days date, press control and at the same time. Times default to a 24 hour clock. Use a or p to indicate am or pm if you use a 12 hour clock (e. g. 8:30 p is interpreted as 8:30 PM). To enter the current time, press control and : (shift-semicolon) at the same time. An entry interpreted as a value (number, date, or time) is aligned to the right side of the cell, to reformat a value. Rounding Numbers that Meet Specified Criteria: To apply colors to maximum andor minimum values: Select a cell in the region, and press CtrlShift (in Excel 2003, press this or CtrlA) to select the Current Region. From the Format menu, select Conditional Formatting. In Condition 1, select Formula Is, and type MAX(F:F) F1. Click Format, select the Font tab, select a color, and then click OK. In Condition 2, select Formula Is, and type MIN(F:F) F1. Repeat step 4, select a different color than you selected for Condition 1, and then click OK. Note: Be sure to distinguish between absolute reference and relative reference when entering the formulas. Rounding Numbers that Meet Specified Criteria Problem: Rounding all the numbers in column A to zero decimal places, except for those that have 5 in the first decimal place. Solution: Use the IF, MOD, and ROUND functions in the following formula: IF(MOD(A2,1)0.5,A2,ROUND(A2,0)) To Copy and Paste All Cells in a Sheet Select the cells in the sheet by pressing CtrlA (in Excel 2003, select a cell in a blank area before pressing CtrlA, or from a selected cell in a Current RegionList range, press CtrlAA). OR Click Select All at the top-left intersection of rows and columns. Press CtrlC. Press CtrlPage Down to select another sheet, then select cell A1. Press Enter. To Copy the Entire Sheet Copying the entire sheet means copying the cells, the page setup parameters, and the defined range Names. Option 1: Move the mouse pointer to a sheet tab. Press Ctrl, and hold the mouse to drag the sheet to a different location. Release the mouse button and the Ctrl key. Option 2: Right-click the appropriate sheet tab. From the shortcut menu, select Move or Copy. The Move or Copy dialog box enables one to copy the sheet either to a different location in the current workbook or to a different workbook. Be sure to mark the Create a copy checkbox. Option 3: From the Window menu, select Arrange. Select Tiled to tile all open workbooks in the window. Use Option 1 (dragging the sheet while pressing Ctrl) to copy or move a sheet. Sorting by Columns The default setting for sorting in Ascending or Descending order is by row. To sort by columns: From the Data menu, select Sort, and then Options. Select the Sort left to right option button and click OK. In the Sort by option of the Sort dialog box, select the row number by which the columns will be sorted and click OK. Descriptive Statistics The Data Analysis ToolPak has a Descriptive Statistics tool that provides you with an easy way to calculate summary statistics for a set of sample data. Summary statistics includes Mean, Standard Error, Median, Mode, Standard Deviation, Variance, Kurtosis, Skewness, Range, Minimum, Maximum, Sum, and Count. This tool eliminates the need to type indivividual functions to find each of these results. Excel includes elaborate and customisable toolbars, for example the standard toolbar shown here: Some of the icons are useful mathematical computation: is the Autosum icon, which enters the formula sum() to add up a range of cells. is the FunctionWizard icon, which gives you access to all the functions available. is the GraphWizard icon, giving access to all graph types available, as shown in this display: Excel can be used to generate measures of location and variability for a variable. Suppose we wish to find descriptive statistics for a sample data: 2, 4, 6, and 8. Step 1. Select the Tools pull-down menu, if you see data analysis, click on this option, otherwise, click on add-in. option to install analysis tool pak. Step 2. Click on the data analysis option. Step 3. Choose Descriptive Statistics from Analysis Tools list. Step 4. When the dialog box appears: Enter A1:A4 in the input range box, A1 is a value in column A and row 1 . in this case this value is 2. Using the same technique enter other VALUES until you reach the last one. If a sample consists of 20 numbers, you can select for example A1, A2, A3, etc. as the input range. Step 5. Select an output range . in this case B1. Click on summary statistics to see the results. When you click OK . you will see the result in the selected range. As you will see, the mean of the sample is 5, the median is 5, the standard deviation is 2.581989, the sample variance is 6.666667,the range is 6 and so on. Each of these factors might be important in your calculation of different statistical procedures. Normal Distribution Consider the problem of finding the probability of getting less than a certain value under any normal probability distribution. As an illustrative example, let us suppose the SAT scores nationwide are normally distributed with a mean and standard deviation of 500 and 100, respectively. Answer the following questions based on the given information: A: What is the probability that a randomly selected student score will be less than 600 points B: What is the probability that a randomly selected student score will exceed 600 points C: What is the probability that a randomly selected student score will be between 400 and 600 Hint: Using Excel you can find the probability of getting a value approximately less than or equal to a given value. In a problem, when the mean and the standard deviation of the population are given, you have to use common sense to find different probabilities based on the question since you know the area under a normal curve is 1. In the work sheet, select the cell where you want the answer to appear. Suppose, you chose cell number one, A1. From the menus, select quotinsert pull-downquot. Steps 2-3 From the menus, select insert, then click on the Function option. Step 4. After clicking on the Function option, the Paste Function dialog appears from Function Category. Choose Statistical then NORMDIST from the Function Name box Click OK Step 5. After clicking on OK, the NORMDIST distribution box appears: i. Enter 600 in X (the value box) ii. Enter 500 in the Mean box iii. Enter 100 in the Standard deviation box iv. Type quottruequot in the cumulative box, then click OK. As you see the value 0.84134474 appears in A1, indicating the probability that a randomly selected students score is below 600 points. Using common sense we can answer part quotbquot by subtracting 0.84134474 from 1. So the part quotbquot answer is 1- 0.8413474 or 0.158653. This is the probability that a randomly selected students score is greater than 600 points. To answer part quotcquot, use the same techniques to find the probabilities or area in the left sides of values 600 and 400. Since these areas or probabilities overlap each other to answer the question you should subtract the smaller probability from the larger probability. The answer equals 0.84134474 - 0.15865526 that is, 0.68269. The screen shot should look like following: Calculating the value of a random variable often called the quotxquot value You can use NORMINV from the function box to calculate a value for the random variable - if the probability to the left side of this variable is given. Actually, you should use this function to calculate different percentiles. In this problem one could ask what is the score of a student whose percentile is 90 This means approximately 90 of students scores are less than this number. On the other hand if we were asked to do this problem by hand, we would have had to calculate the x value using the normal distribution formula x m zd. Now lets use Excel to calculate P90. In the Paste function, dialog click on statistical, then click on NORMINV . The screen shot would look like the following: When you see NORMINV the dialog box appears. Eu. Enter 0.90 for the probability (this means that approximately 90 of students score is less than the value we are looking for) ii. Enter 500 for the mean (this is the mean of the normal distribution in our case) iii. Enter 100 for the standard deviation (this is the standard deviation of the normal distribution in our case) At the end of this screen you will see the formula result which is approximately 628 points. This means the top 10 of the students scored better than 628. Confidence Interval for the Mean Suppose we wish for estimating a confidence interval for the mean of a population. Depending on the size of your sample size you may use one of the following cases: Large Sample Size (n is larger than, say 30): The general formula for developing a confidence interval for a population means is: In this formula is the mean of the sample Z is the interval coefficient, which can be found from the normal distribution table (for example the interval coefficient for a 95 confidence level is 1.96). S is the standard deviation of the sample and n is the sample size. Now we would like to show how Excel is used to develop a certain confidence interval of a population mean based on a sample information. As you see in order to evaluate this formula you need quotthe mean of the samplequot and the margin of error Excel will automatically calculate these quantities for you. The only things you have to do are: add the margin of error to the mean of the sample, Find the upper limit of the interval and subtract the margin of error from the mean to the lower limit of the interval. To demonstrate how Excel finds these quantities we will use the data set, which contains the hourly income of 36 work-study students here, at the University of Baltimore. These numbers appear in cells A1 to A36 on an Excel work sheet. After entering the data, we followed the descriptive statistic procedure to calculate the unknown quantities. The only additional step is to click on the confidence interval in the descriptive statistics dialog box and enter the given confidence level, in this case 95. Here is, the above procedures in step-by-step: Step 1. Enter data in cells A1 to A36 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option then click OK . On the descriptive statistics dialog, click on Summary Statistic. After you have done that, click on the confidence interval level and type 95 - or in other problems whatever confidence interval you desire. In the Output Range box enter B1 or what ever location you desire. Now click on OK . The screen shot would look like the following: As you see, the spreadsheet shows that the mean of the sample is 6.902777778 and the absolute value of the margin of error 0.231678109. This mean is based on this sample information. A 95 confidence interval for the hourly income of the UB work-study students has an upper limit of 6.902777778 0.231678109 and a lower limit of 6.902777778 - 0.231678109. On the other hand, we can say that of all the intervals formed this way 95 contains the mean of the population. Or, for practical purposes, we can be 95 confident that the mean of the population is between 6.902777778 - 0.231678109 and 6.902777778 0.231678109. We can be at least 95 confident that interval 6.68 and 7.13 contains the average hourly income of a work-study student. Smal Sample Size (say less than 30) If the sample n is less than 30 or we must use the small sample procedure to develop a confidence interval for the mean of a population. The general formula for developing confidence intervals for the population mean based on small a sample is: In this formula is the mean of the sample. is the interval coefficient providing an area of in the upper tail of a t distribution with n-1 degrees of freedom which can be found from a t distribution table (for example the interval coefficient for a 90 confidence level is 1.833 if the sample is 10). S is the standard deviation of the sample and n is the sample size. Now you would like to see how Excel is used to develop a certain confidence interval of a population mean based on this small sample information. As you see, to evaluate this formula you need quotthe mean of the samplequot and the margin of error Excel will automatically calculate these quantities the way it did for large samples. Again, the only things you have to do are: add the margin of error to the mean of the sample, , find the upper limit of the interval and to subtract the margin of error from the mean to find the lower limit of the interval. To demonstrate how Excel finds these quantities we will use the data set, which contains the hourly incomes of 10 work-study students here, at the University of Baltimore. These numbers appear in cells A1 to A10 on an Excel work sheet. After entering the data we follow the descriptive statistic procedure to calculate the unknown quantities (exactly the way we found quantities for large sample). Here you are with the procedures in step-by-step form: Step 1. Enter data in cells A1 to A10 on the spreadsheet Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option. Click OK on the descriptive statistics dialog, click on Summary Statistic, click on the confidence interval level and type in 90 or in other problems whichever confidence interval you desire. In the Output Range box, enter B1 or whatever location you desire. Now click on OK . The screen shot will look like the following: Now, like the calculation of the confidence interval for the large sample, calculate the confidence interval of the population based on this small sample information. The confidence interval is: 6.8 0.414426102 or 6.39 7.21. We can be at least 90 confidant that the interval 6.39 and 7.21 contains the true mean of the population. Test of Hypothesis Concerning the Population Mean Again, we must distinguish two cases with respect to the size of your sample Large Sample Size (say, over 30): In this section you wish to know how Excel can be used to conduct a hypothesis test about a population mean. We will use the hourly incomes of different work-study students than those introduced earlier in the confidence interval section. Data are entered in cells A1 to A36. The objective is to test the following Null and Alternative hypothesis: The null hypothesis indicates that the average hourly income of a work-study student is equal to 7 per hour however, the alternative hypothesis indicates that the average hourly income is not equal to 7 per hour. I will repeat the steps taken in descriptive statistics and at the very end will show how to find the value of the test statistics in this case, z, using a cell formula. Step 1. Enter data in cells A1 to A36 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option, click OK . On the descriptive statistics dialog, click on Summary Statistic. Select the Output Range box, enter B1 or whichever location you desire. Agora clique em OK. (To calculate the value of the test statistics search for the mean of the sample then the standard error. In this output, these values are in cells C3 and C4.) Step 4. Select cell D1 and enter the cell formula (C3 - 7)C4. The screen shot should look like the following: The value in cell D1 is the value of the test statistics. Since this value falls in acceptance range of -1.96 to 1.96 (from the normal distribution table), we fail to reject the null hypothesis. Small Sample Size (say, less than 30): Using steps taken the large sample size case, Excel can be used to conduct a hypothesis for small-sample case. Lets use the hourly income of 10 work-study students at UB to conduct the following hypothesis. The null hypothesis indicates that average hourly income of a work-study student is equal to 7 per hour. The alternative hypothesis indicates that average hourly income is not equal to 7 per hour. I will repeat the steps taken in descriptive statistics and at the very end will show how to find the value of the test statistics in this case quottquot using a cell formula. Step 1. Enter data in cells A1 to A10 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option. Clique em OK. On the descriptive statistics dialog, click on Summary Statistic. Select the Output Range boxes, enter B1 or whatever location you chose. Again, click on OK . (To calculate the value of the test statistics search for the mean of the sample then the standard error, in this output these values are in cells C3 and C4.) Step 4. Select cell D1 and enter the cell formula (C3 - 7)C4. The screen shot would look like the following: Since the value of test statistic t -0.66896 falls in acceptance range -2.262 to 2.262 (from t table, where 0.025 and the degrees of freedom is 9), we fail to reject the null hypothesis. Difference Between Mean of Two Populations In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means assuming that populations have equal variances. The data in this case are taken from various offices here at the University of Baltimore. I collected the hourly income data of 36 randomly selected work-study students and 36 student assistants. The hourly income range for work-study students was 6 - 8 while the hourly income range for student assistants was 6-9. The main objective in this hypothesis testing is to see whether there is a significant difference between the means of the two populations. The NULL and the ALTERNATIVE hypothesis is that the means are equal and the means are not equal, respectively. Referring to the spreadsheet, I chose A1 and A2 as label centers. The work-study students hourly income for a sample size 36 are shown in cells A2:A37 . and the student assistants hourly income for a sample size 36 is shown in cells B2:B37 Data for Work Study Student: 6, 6, 6, 6, 6, 6, 6, 6.5, 6.5, 6.5, 6.5, 6.5, 6.5, 7, 7, 7, 7, 7, 7, 7, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 8, 8, 8, 8, 8, 8, 8, 8, 8. Data for Student Assistant: 6, 6, 6, 6, 6, 6.5, 6.5, 6.5, 6.5, 6.5, 7, 7, 7, 7, 7, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 8, 8, 8, 8, 8, 8, 8, 8.5, 8.5, 8.5, 8.5, 8.5, 9, 9, 9, 9. Use the Descriptive Statistics procedure to calculate the variances of the two samples. The Excel procedure for testing the difference between the two population means will require information on the variances of the two populations. Since the variances of the two populations are unknowns they should be replaced with sample variances. The descriptive for both samples show that the variance of first sample is s 1 2 0.55546218 . while the variance of the second sample s 2 2 0.969748 . To conduct the desired test hypothesis with Excel the following steps can be taken: Step 1. From the menus select Tools then click on the Data Analysis option. Step 2. When the Data Analysis dialog box appears: Choose z-Test: Two Sample for means then click OK Step 3. When the z-Test: Two Sample for means dialog box appears: Enter A1:A36 in the variable 1 range box (work-study students hourly income) Enter B1:B36 in the variable 2 range box (student assistants hourly income) Enter 0 in the Hypothesis Mean Difference box (if you desire to test a mean difference other than 0, enter that value) Enter the variance of the first sample in the Variable 1 Variance box Enter the variance of the second sample in the Variable 2 Variance box and select Labels Enter 0.05 or, whatever level of significance you desire, in the Alpha box Select a suitable Output Range for the results, I chose C19 . then click OK. The value of test statistic z-1.9845824 appears in our case in cell D24. The rejection rule for this test is z 1.96 from the normal distribution table. In the Excel output these values for a two-tail test are z 1.959961082. Since the value of the test statistic z-1.9845824 is less than -1.959961082 we reject the null hypothesis. We can also draw this conclusion by comparing the p-value for a two tail - test and the alpha value. Since p-value 0.047190813 is less than a0.05 we reject the null hypothesis. Overall we can say, based on the sample results, the two populations means are different. Small Samples: n 1 OR n 2 are less than 30 In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means. - Given that the populations have equal variances when two small independent samples are taken from both populations. Similar to the above case, the data in this case are taken from various offices here at the University of Baltimore. I collected hourly income data of 11 randomly selected work-study students and 11 randomly selected student assistants. The hourly income range for both groups was similar range, 6 - 8 and 6-9. The main objective in this hypothesis testing is similar too, to see whether there is a significant difference between the means of the two populations. The NULL and the ALTERNATIVE hypothesis are that the means are equal and they are not equal, respectively. Referring to the spreadsheet, we chose A1 and A2 as label centers. The work-study students hourly income for a sample size 11 are shown in cells A2:A12 . and the student assistants hourly income for a sample size 11 is shown in cells B2:B12 . Unlike previous case, you do not have to calculate the variances of the two samples, Excel will automatically calculate these quantities and use them in the calculation of the value of the test statistic. Similar to the previous case, but a bit different in step 2, to conduct the desired test hypothesis with Excel the following steps can be taken: Step 1. From the menus select Tools then click on the Data Analysis option. Step 2. When the Data Analysis dialog box appears: Choose t-Test: Two Sample Assuming Equal Variances then click OK Step 3 When the t-Test: Two Sample Assuming Equal Variances dialog box appears : Enter A1:A12 in the variable 1 range box (work-study student hourly income) Enter B1:B12 in the variable 2 range box (student assistant hourly income) Enter 0 in the Hypothesis Mean Difference box(if you desire to test a mean difference other than zero, enter that value) then select Labels Enter 0.05 or, whatever level of significance you desire, in the Alpha box Select a suitable Output Range for the results, I chose C1, then click OK. The value of the test statistic t-1.362229828 appears, in our case, in cell D10. The rejection rule for this test is t 2.086 from the t distribution table where the t value is based on a t distribution with n 1 - n 2 -2 degrees of freedom and where the area of the upper one tail is 0.025 ( that is equal to alpha2). In the Excel output the values for a two-tail test are t 2.085962478. Since the value of the test statistic t-1.362229828, is in an acceptance range of t 2.085962478, we fail to reject the null hypothesis. We can also draw this conclusion by comparing the p-value for a two-tail test and the alpha value. Since the p-value 0.188271278 is greater than a0.05 again . we fail to reject the null hypothesis. Overall we can say, based on sample results, the two populations means are equal. Enter data in an Excel work sheet starting with cell A2 and ending with cell C8. The following steps should be taken to find the proper output for interpretation. Step 1. From the menus select Tools and click on Data Analysis option. Step 2. When data analysis dialog appears, choose Anova single-factor option enter A2:C8 in the input range box. Select labels in first row. Step3. Select any cell as output(in here we selected A11). Click OK. The general form of Anova table looks like following: Source of Variation Suppose the test is done at level of significance a 0.05, we reject the null hypothesis. This means there is a significant difference between means of hourly incomes of student assistants in these departments. The Two-way ANOVA Without Replication In this section, the study involves six students who were offered different hourly wages in three different department services here at the University of Baltimore. The objective is to see whether the hourly incomes are the same. Therefore, we can consider the following: Treatment: Hourly payments in the three departments Blocks: Each student is a block since each student has worked in the three different departments The general form of Anova table would look like: Source of Variation Degrees of freedom To find the Excel output for the above data the following steps can be taken: Step 1. From the menus select Tools and click on Data Analysis option. Step2. When data analysis box appears: select Anova two-factor without replication then Enter A2: D8 in the input range. Select labels in first row. Step3. Select an output range (in here we selected A11) then OK. Source of Variation NOTE: FMSTMSE 0.9805560.497222 1.972067 F 3.33 from table (5 numerator DF and 10 denominator DF) Since 1.972067 Goodness-of-Fit Test for Discrete Random Variables The CHI-SQUARE distribution can be used in a hypothesis test involving a population variance. However, in this section we would like to test and see how close a sample results are to the expected results. Example: The Multinomial Random Variable In this example the objective is to see whether or not based on a randomly selected sample information the standards set for a population is met. There are so many practical examples that can be used in this situation. For example it is assumed the guidelines for hiring people with different ethnic background for the US government is set at 70(WHITE), 20(African American) and 10(others), respectively. A randomly selected sample of 1000 US employees shows the following results that is summarized in a table. EXPECTED NUMBER OF EMPLOYEES OBSERVED FROM SAMPLE As you see the observed sample numbers for groups two and three are lower than their expected values unlike group one which has a higher expected value. Is this a clear sign of discrimination with respect to ethnic background Well depends on how much lower the expected values are. The lower amount might not statistically be significant. To see whether these differences are significant we can use Excel and find the value of the CHI-SQUARE. If this value falls within the acceptance region we can assume that the guidelines are met otherwise they are not. Now lets enter these numbers into Excel spread - sheet. We used cells B7-B9 for the expected proportions, C7-C9 for the observed values and D7-D9 for the expected frequency. To calculate the expected frequency for a category, you can multiply the proportion of that category by the sample size (in here 1000). The formula for the first cell of the expected value column, D7 is 1000B7. To find other entries in the expected value column, use the copy and the paste menu as shown in the following picture. These are important values for the chi-square test. The observed range in this case is C7: C9 while the expected range is D7: D9. The null and the alternative hypothesis for this test are as follows: H A . The population proportions are not P W 0.70, P A 0.20 and P O 0.10 Now lets use Excel to calculate the p-value in a CHI-SQUARE test. Step 1. Select a cell in the work sheet, the location which you like the p value of the CHI-SQUARE to appear. We chose cell D12. Step 2. From the menus, select insert then click on the Function option, Paste Function dialog box appears. Step 3. Refer to function category box and choose statistical . from function name box select CHITEST and click on OK . Step 4. When the CHITEST dialog appears: Enter C7: C9 in the actual-range box then enter D7: D9 in the expected-range box, and finally click on OK . The p-value will appear in the selected cell, D12. As you see the p value is 0.002392 which is less than the value of the level of significance (in this case the level of significance, a 0.10). Hence the null hypothesis should be rejected. This means based on the sample information the guidelines are not met. Notice if you type CHITEST(C7:C9,D7:D9) in the formula bar the p-value will show up in the designated cell. NOTE: Excel can actually find the value of the CHI-SQUARE. To find this value first select an empty cell on the spread sheet then in the formula bar type CHIINV(D12,2). D12 designates the p-Value found previously and 2 is the degrees of freedom (number of rows minus one). The CHI-SQUARE value in this case is 12.07121. If we refer to the CHI-SQUARE table we will see that the cut off is 4.60517 since 12.071214.60517 we reject the null. The following screen shot shows you how to the CHI-SQUARE value. Test of Independence: Contingency Tables The CHI-SQUARE distribution is also used to test and see whether two variables are independent or not. For example based on sample data you might want to see whether smoking and gender are independent events for a certain population. The variables of interest in this case are smoking and the gender of an individual. Another example in this situation could involve the age range of an individual and his or her smoking habit. Similar to case one data may appear in a table but unlike the case one this table may contains several columns in addition to rows. The initial table contains the observed values. To find expected values for this table we set up another table similar to this one. To find the value of each cell in the new table we should multiply the sum of the cell column by the sum of the cell row and divide the results by the grand total. The grand total is the total number of observations in a study. Now based on the following table test whether or not the smoking habit and gender of the population that the following sample taken from are independent. On the other hand is that true that males in this population smoke more than females You could use formula bar to calculate the expected values for the expected range. For example to find the expected value for the cell C5 which is replaced in c11 you could click on the formula bar and enter C6D5D6 then enter in cell C11. Step 1. Observed Range b4:c5 Smoking and gender So the observed range is b4:c5 and the expected range is b10:c11. Step 3. Click on fx (paste function) Step 4. When Paste Function dialog box appears, click on Statistical in function category and CHITEST in the function name then click OK. When the CHITEST box appears, enter b4:c5 for the actual range, then b10:c11 for the expected range. Step 5. Click on OK (the p-value appears). 0.477395 Conclusion: Since p-value is greater than the level of significance (0.05), fails to reject the null. This means smoking and gender are independent events. Based on sample information one can not assure females smoke more than males or the other way around. Step 6. To find the chi-square value, use CHINV function, when Chinv box appears enter 0.477395 for probability part, then 1 for the degrees of freedom. Degrees of freedom(number of columns-1)X(number of rows-1) Test Hypothesis Concerning the Variance of Two Populations In this section we would like to examine whether or not the variances of two populations are equal. Whenever independent simple random samples of equal or different sizes such as n 1 and n 2 are taken from two normal distributions with equal variances, the sampling distribution of s 1 2 s 2 2 has F distribution with n 1 - 1 degrees of freedom for the numerator and n 2 - 1 degrees of freedom for the denominator. In the ratio s 1 2 s 2 2 the numerator s 1 2 and the denominator s 2 2 are variances of the first and the second sample, respectively. The following figure shows the graph of an F distribution with 10 degrees of freedom for both the numerator and the denominator. Unlike the normal distribution as you see the F distribution is not symmetric. The shape of an F distribution is positively skewed and depends on the degrees of freedom for the numerator and the denominator. The value of F is always positive. Now let see whether or not the variances of hourly income of student-assistant and work-study students based on samples taken from populations previously are equal. Assume that the hypothesis test in this case is conducted at a 0.10. The null and the alternative are: Rejection Rule: Reject the null hypothesis if Flt F 0.095 or Fgt F 0.05 where F, the value of the test statistic is equal to s 1 2 s 2 2. with 10 degrees of freedom for both the numerator and the denominator. We can find the value of F .05 from the F distribution table. If s 1 2 s 2 2. we do not need to know the value of F 0.095 otherwise, F 0.95 1 F 0.05 for equal sample sizes. A survey of eleven student-assistant and eleven work-study students shows the following descriptive statistics. Our objective is to find the value of s 1 2 s 2 2. where s 1 2 is the value of the variance of student assistant sample and s 2 2 is the value of the variance of the work study students sample. As you see these values are in cells F8 and D8 of the descriptive statistic output. To calculate the value of s 1 2 s 2 2. select a cell such as A16 and enter cell formula F8D8 and enter. This is the value of F in our problem. Since this value, F1.984615385, falls in acceptance area we fail to reject the null hypothesis. Hence, the sample results do support the conclusion that student assistants hourly income variance is equal to the work study students hourly income variance. The following screen shoot shows how to find the F value. We can follow the same format for one tail test(s). Linear Correlation and Regression Analysis In this section the objective is to see whether there is a correlation between two variables and to find a model that predicts one variable in terms of the other variable. There are so many examples that we could mention but we will mention the popular ones in the world of business. Usually independent variable is presented by the letter x and the dependent variable is presented by the letter y. A business man would like to see whether there is a relationship between the number of cases of sold and the temperature in a hot summer day based on information taken from the past. He also would like to estimate the number cases of soda which will be sold in a particular hot summer day in a ball game. He clearly recorded temperatures and number of cases of soda sold on those particular days. The following table shows the recorded data from June 1 through June 13. The weatherman predicts a 94F degree temperature for June 14. The businessman would like to meet all demands for the cases of sodas ordered by customers on June 14. Now lets use Excel to find the linear correlation coefficient and the regression line equation. The linear correlation coefficient is a quantity between -1 and 1. This quantity is denoted by R . The closer R to 1 the stronger positive (direct) correlation and similarly the closer R to -1 the stronger negative (inverse) correlation exists between the two variables. The general form of the regression line is y mx b. In this formula, m is the slope of the line and b is the y-intercept. You can find these quantities from the Excel output. In this situation the variable y (the dependent variable) is the number of cases of soda and the x (independent variable) is the temperature. To find the Excel output the following steps can be taken: Step 1. From the menus choose Tools and click on Data Analysis. Step 2. When Data Analysis dialog box appears, click on correlation. Step 3. When correlation dialog box appears, enter B1:C14 in the input range box. Click on Labels in first row and enter a16 in the output range box. Click on OK. As you see the correlation between the number of cases of soda demanded and the temperature is a very strong positive correlation. This means as the temperature increases the demand for cases of soda is also increasing. The linear correlation coefficient is 0.966598577 which is very close to 1. Now lets follow same steps but a bit different to find the regression equation. Step 1. From the menus choose Tools and click on Data Analysis Step 2 . When Data Analysis dialog box appears, click on regression . Step 3. When Regression dialog box appears, enter b1:b14 in the y-range box and c1:c14 in the x-range box. Click on labels . Step 4. Enter a19 in the output range box . Note: The regression equation in general should look like Ym X b. In this equation m is the slope of the regression line and b is its y-intercept. Adjusted R Square The relationship between the number of cans of soda and the temperature is: Y 0.879202711 X 9.17800767 The number of cans of soda 0.879202711(Temperature) 9.17800767. Referring to this expression we can approximately predict the number of cases of soda needed on June 14. The weather forecast for this is 94 degrees, hence the number of cans of soda needed is equal to The number of cases of soda0.879202711(94) 9.17800767 91.82 or about 92 cases. Moving Average and Exponential Smoothing Moving Average Models: Use the Add Trendline option to analyze a moving average forecasting model in Excel. You must first create a graph of the time series you want to analyze. Select the range that contains your data and make a scatter plot of the data. Once the chart is created, follow these steps: Click on the chart to select it, and click on any point on the line to select the data series. When you click on the chart to select it, a new option, Chart, s added to the menu bar. From the Chart menu, select Add Trendline. The following is the moving average of order 4 for weekly sales: Exponential Smoothing Models: The simplest way to analyze a timer series using an Exponential Smoothing model in Excel is to use the data analysis tool. This tool works almost exactly like the one for Moving Average, except that you will need to input the value of a instead of the number of periods, k. Once you have entered the data range and the damping factor, 1- a. and indicated what output you want and a location, the analysis is the same as the one for the Moving Average model. Applications and Numerical Examples Descriptive Statistics: Suppose you have the following, n 10, data: 1.2, 1.5, 2.6, 3.8, 2.4, 1.9, 3.5, 2.5, 2.4, 3.0 Type your n data points into the cells A1 through An. Click on the Tools menu. (At the bottom of the Tools menu will be a submenu Data Analysis. , if the Analysis Tool Pack has been properly installed.) Clicking on Data Analysis. will lead to a menu from which Descriptive Statistics is to be selected. Select Descriptive Statistics by pointing at it and clicking twice, or by highlighting it and clicking on the Okay button. Within the Descriptive Statistics submenu, a. for the input range enter A1:Dn, assuming you typed the data into cells A1 to An. b. click on the output range button and enter the output range C1:C16. C. click on the Summary Statistics box d. finally, click on Okay. The Central Tendency: The data can be sorted in ascending order: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 The mean, median and mode are computed as follows: (1.2 1.5 2.6 3.8 2.4 1.9 3.5 2.5 2.4 3.0) 10 2.48 The mode is 2.4, since it is the only value that occurs twice. The midrange is (1.2 3.8) 2 2.5. Note that the mean, median and mode of this set of data are very close to each other. This suggests that the data is very symmetrically distributed. Variance: The variance of a set of data is the average of the cumulative measure of the squares of the difference of all the data values from the mean. The sample variance-based estimation for the population variance are computed differently. The sample variance is simply the arithmetic mean of the squares of the difference between each data value in the sample and the mean of the sample. On the other hand, the formula for an estimate for the variance in the population is similar to the formula for the sample variance, except that the denominator in the fraction is (n-1) instead of n. However, you should not worry about this difference if the sample size is large, say over 30. Compute an estimate for the variance of the population . given the following sorted data: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 mean 2.48 as computed earlier. An estimate for the population variance is: s 2 1 (10-1) (1.2 - 2.48) 2 (1.5 - 2.48) 2 (1.9 - 2.48) 2 (2.4 -2.48) 2 (2.4 - 2.48) 2 (2.5 - 2.48) 2 (2.6 - 2.48) 2 (3.0 - 2.48) 2 (3.5 -2.48) 2 (3.8 - 2.48) 2 (1 9) (1.6384 0.9604 0.3364 0.0064 0.0064 0.0004 0.0144 0.2704 1.0404 1.7424) 0.6684 Therefore, the standard deviation is s ( 0.6684 ) 12 0.8176 Probability and Expected Values: Newsweek reported that average take for bank robberies was 3,244 but 85 percent of the robbers were caught. Assuming 60 percent of those caught lose their entire take and 40 percent lose half, graph the probability mass function using EXCEL. Calculate the expected take from a bank robbery. Does it pay to be a bank robber To construct the probability function for bank robberies, first define the random variable x, bank robbery take. If the robber is not caught, x 3,244. If the robber is caught and manages to keep half, x 1,622. If the robber is caught and loses it all, then x 0. The associated probabilities for these x values are 0.15 (1 - 0.85), 0.34 (0.85)(0.4), and 0.51 (0.85)(0.6). After entering the x values in cells A1, A2 and A3 and after entering the associated probabilities in B1, B2, and B3, the following steps lead to the probability mass function: Click on ChartWizard. The ChartWizard Step 1 of 4 screen will appear. Highlight Column at ChartWizard Step 1 of 4 and click Next. At ChartWizard Step 2 of 4 Chart Source Data, enter B1:B3 for Data range, and click column button for Series in. A graph will appear. Click on series toward the top of the screen to get a new page. At the bottom of the Series page, is a rectangle for Category (X) axis labels: Click on this rectangle and then highlight A1:A3. At Step 3 of 4 move on by clicking on Next, and at Step 4 of 4, click on Finish. The expected value of a robbery is 1,038.08. E(X) (0)(0.51)(1622)(0.34) (3244)(0.15) 0 551.48 486.60 1038.08 The expected return on a bank robbery is positive. On average, bank robbers get 1,038.08 per heist. If criminals make their decisions strictly on this expected value, then it pays to rob banks. A decision rule based only on an expected value, however, ignores the risks or variability in the returns. In addition, our expected value calculations do not include the cost of jail time, which could be viewed by criminals as substantial. Discrete Continuous Random Variables: Binomial Distribution Application: A multiple choice test has four unrelated questions. Each question has five possible choices but only one is correct. Thus, a person who guesses randomly has a probability of 0.2 of guessing correctly. Draw a tree diagram showing the different ways in which a test taker could get 0, 1, 2, 3 and 4 correct answers. Sketch the probability mass function for this test. What is the probability a person who guesses will get two or more correct Solution: Letting Y stand for a correct answer and N a wrong answer, where the probability of Y is 0.2 and the probability of N is 0.8 for each of the four questions, the probability tree diagram is shown in the textbook on page 182. This probability tree diagram shows the branches that must be followed to show the calculations captured in the binomial mass function for n 4 and 0.2. For example, the tree diagram shows the six different branch systems that yield two correct and two wrong answers (which corresponds to 4(22) 6. The binomial mass function shows the probability of two correct answers as P(x 2 n 4, p 0.2) 6(.2)2(.8)2 6(0.0256) 0.1536 P(2) Which is obtained from excel by using the BINOMDIST Command, where the first entry is x, the second is n, and the third is mass (0) or cumulative (1) that is, entering BINOMDIST(2,4,0.2,0) IN ANY EXCEL CELL YIELDS 0.1536 AND BINOMDIST(3,4,0.2,0) YIELDS P(x3n4, p 0.2) 0.0256 BINOMDIST(4,4,0.2,0) YIELDS P(x4n4, p 0.2) 0.0016 1-BINOMDIST(1,4,0.2,1) YIELDS P(x 179 2 n 4, p 0.2) 0.1808 Normal Example: If the time required to complete an examination by those with a certain learning disability is believed to be distributed normally, with mean of 65 minutes and a standard deviation of 15 minutes, then when can the exam be terminated so that 99 percent of those with the disability can finish Solution: Because t he average and standard deviation are known, what needs to be established is the amount of time, above the mean time, such that 99 percent of the distribution is lower. This is a distance that is measured in standard deviations as given by the Z value corresponding to the 0.99 probability found in the body of Appendix B, Table 5,as shown in the textbook OR the commands entered into any cell of Excel to find this Z value is NORMINV(0.99,0,1) for 2.326342. The closest cumulative probability that can be found is 0.9901, in the row labeled 2.3 and column headed by .03, Z 2.33, which is only an approximation for the more exact 2.326342 found in Excel. Using this more exact value the calculation with mean m and standard deviation s in the following formula would be Z ( X - m ) s That is, Z ( x - 65)15 Thus, x 65 15(2.32634) 99.9 minutes. Alternatively, instead of standardizing with the Z distribution using Excel we can simply work directly with the normal distribution with a mean of 65 and standard deviation of 15 and enter NORMINV(0.99,65,15). In general to obtain the x value for which alpha percent of a normal random variables values are lower, the following NORMINV command may be used, where the first entry is a. the second is m. and the third is s. Another Example: In the early 1980s, the Toro Company of Minneapolis, Minnesota, advertised that it would refund the purchase price of a snow blower if the following winters snowfall was less than 21 percent of the local average. If the average snowfall is 45.25 inches, with a standard deviation of 12.2 inches, what is the likelihood that Toro will have to make refunds Solution: Within limits, snowfall is a continuous random variable that can be expected to vary symmetrically around its mean, with values closer to the mean occurring most often. Thus, it seems reasonable to assume that snowfall (x) is approximately normally distributed with a mean of 45.25 inches and standard deviation of 12.2 inches. Nine and one half inches is 21 percent of the mean snowfall of 45.25 inches and, with a standard deviation of 12.2 inches, the number of standard deviations between 45.25 inches and 9.5 inches is Z: Z ( x - m ) s (9.50 - 45.25)12.2 -2.93 Using Appendix B, Table 5, the textbook demonstrates the determination of P(x 163 9.50) P(z 163 -2.93) 0.17, the probability of snowfall less than 9.5 inches. Using Excel, this normal probability is obtained with the NORMDIST command, where the first entry is x, the second is mean m. the third is standard deviation s, and the fourth is CUMULATIVE (1). Entering NORMDIST(9.5,45.25,12.2,1), Gives P( x 163 9.50) 0.001693. Sampling Distribution and the Central Limit Theorem : A bakery sells an average of 24 loaves of bread per day. Sales (x) are normally distributed with a standard deviation of 4. If a random sample of size n 1 (day) is selected, what is the probability this x value will exceed 28 If a random sample of size n 4 (days) is selected, what is theprobability that xbar 179 28 Why does the answer in part 1 differ from that in part 2 1. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 4. Thus, using Excel, 0.15866 1-NORMDIST(28,24,4,1). 2. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 2 using Excel, 0.02275 1-NORMDIST(28,24,2,1). Regression Analysis: The highway deaths per 100 million vehicle miles and highway speed limits for 10 countries, are given below: (Death, Speed) (3.0, 55), (3.3, 55), (3.4, 55), (3.5, 70), (4.1, 55), (4.3, 60), (4.7, 55), (4.9, 60), (5.1, 60), and (6.1, 75). From this we can see that five countries with the same speed limit have very different positions on the safety list. For example, Britain. with a speed limit of 70 is demonstrably safer than Japan, at 55. Can we argue that, speed has little to do with safety. Use regression analysis to answer this question. Solution: Enter the ten paired y and x data into cells A2 to A11 and B2 to B11, with the death rate label in A1 and speed limits label in B1, the following steps produce the regression output. Choose Regression from Data Analysis in the Tools menu. The Regression dialog box will will appear. Note: Use the mouse to move between the boxes and buttons. Click on the desired box or button. The large rectangular boxes require a range from the worksheet. A range may be typed in or selected by highlighting the cells with the mouse after clicking on the box. If the dialog box blocks the data, it can be moved on the screen by clicking on the title bar and dragging. For the Input Y Range, enter A1 to A11, and for the Input X Range enter B1 to B11. Because the Y and X ranges include the Death and Speed labels in A1 and B1, select the Labels box with a click. Click the Output Range button and type reference cell, which in this demonstration is A13. To get the predicted values of Y (Death rates) and residuals select the Residuals box with a click. Your screen display should show a Table, clicking OK will give the SUMMARY OUTPUT, ANOVA AND RESIDUAL OUTPUT The first section of the EXCEL printout gives SUMMARY OUTPUT. The Multiple R is the square root of the R Square the computation and interpretation of which we have already discussed. The Standard Error of estimate (which will be discussed in the next chapter) is s 0.86423, which is the square root of Residual SS 5.97511 divided by its degrees of freedom, df 8, as given in the ANOVA section. We will also discuss the adjusted R-square of 0.21325 in the following chapters. Under the ANOVA section are the estimated regression coefficients and related statistics that will be discussed in detail in the next chapter. For now it is sufficient to recognize that the calculated coefficient values for the slope and y intercept are provided (b 0.07556 and a -0.29333). Next to these coefficient estimates is information on the variability in the distribution of the least-squares estimators from which these specific estimates were drawn: the column titled Std. Error contains the standard deviations (standard errors) of the intercept and slope distributions the t-ratio and p columns give the calculated values of the t statistics and associated p-values. As shown in Chapter 13, the t statistic of 1.85458 and p-value of 0.10077, for example, indicates that the sample slope (0.07556) is sufficiently different from zero, at even the 0.10 two-tail Type I error level, to conclude that there is a significant relationship between deaths and speed limits in the population. This conclusion is contrary to assertion that speed has little to do with safety. SUMMARY OUTPUT: Multiple R 0.54833, R Square 0.30067, Adjusted R Square 0.21325, Standard Error 0.86423, Observations 10 ANOVA df SS MS F P-value Regression 1 2.56889 2.56889 3.43945 0.10077 Residual 8 5.97511 0.74689 Total 9 8.54400 Coeffs. Estimate Std. Error T Stat P-value Lower 95 Upper 95 Intercept -0.29333 2.45963 -0.11926 0.90801 -5.96526 5.37860 Speed 0.07556 0.04074 1.85458 0.10077 -0.01839 0.16950 Predicted Residuals 3.86222 -0.86222 3.86222 -0.56222 3.86222 -0.46222 4.99556 -1.49556 3.86222 0.23778 4.24000 0.06000 3.86222 0.83778 4.24000 0.66000 4.24000 0.86000 5.37333 0.72667 Microsoft Excel Add-Ins Forecasting with regression requires the Excel add-in called Analysis ToolPak , and linear programming requires the Excel add-in called Solver . How you check to see if these are activated on your computer, and how to activate them if they are not active, varies with Excel version. Here are instructions for the most common versions. If Excel will not let you activate Data Analysis and Solver, you must use a different computer. Excel 20022003: Start Excel, then click Tools and look for Data Analysis and for Solver. If both are there, press Esc (escape) and continue with the respective assignment. Otherwise click Tools, Add-Ins, and check the boxes for Analysis ToolPak and for Solver, then click OK. Click Tools again, and both tools should be there. Excel 2007: Start Excel 2007 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: - click the 8220Office Button8221 at top left - click the Excel Options button near the bottom of the resulting window - click the Add-ins button on the left of the next screen - near the bottom at Manage Excel Add-ins, click Go - check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK - click the Data tab as above and verify that the add-ins show. Excel 2010: Start Excel 2010 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: - click the File tab at top left - click the Options button near the bottom of the left side - click the Add-ins button near the bottom left of the next screen - near the bottom at Manage Excel Add-ins, click Go - check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK - click the Data tab as above and verify that the add-ins show. Solving Linear Programs by Excel Some of these examples can be modified for other types problems Computer-assisted Learning: E-Labs and Computational Tools My teaching style deprecates the plug the numbers into the software and let the magic box work it out approach. Personal computers, spreadsheets, e. g. Excel. professional statistical packages (e. g. such as SPSS), and other information technologies are now ubiquitous in statistical data analysis. Without using these tools, one cannot perform any realistic statistical data analysis on large data sets. The appearance of other computer software, JavaScript Applets. Statistical Demonstrations Applets. and Online Computation are the most important events in the process of teaching and learning concepts in model-based statistical decision making courses. These tools allow you to construct numerical examples to understand the concepts, and to find their significance for yourself. Use any or online interactive tools available on the WWW to perform statistical experiments (with the same purpose, as you used to do experiments in physics labs to learn physics) to understand statistical concepts such as Central Limit Theorem are entertaining and educating. Computer-assisted learning is similar to the experiential model of learning. The adherents of experiential learning are fairly adamant about how we learn. Learning seldom takes place by rote. Learning occurs because we immerse ourselves in a situation in which we are forced to perform and think. You get feedback from the computer output and then adjust your thinking-process if needed. A SPSS-Example . SPSS-Examples . SPSS-More Examples . (Statistical Package for the Social Sciences) is a data management and analysis product. It can perform a variety of data analysis and presentation functions, including statistical analyses and graphical presentation of data. SAS (Statistical Analysis System) is a system of software packages some of its basic functions and uses are: database management inputting, cleaning and manipulating data, statistical analysis, calculating simple statistics such as means, variances, correlations running standard routines such as regressions. Available at: SPSSSAS Packages on Citrix (Installing and Accessing ) Use your email ID and Password: Technical Difficulties OTS Call Center (401) 837-6262 Excel Examples. Excel More Examples It is Excellent for Descriptive Statistics, and getting acceptance is improving, as computational tool for Inferential Statistics. The Value of Performing Experiment: If the learning environment is focused on background information, knowledge of terms and new concepts, the learner is likely to learn that basic information successfully. However, this basic knowledge may not be sufficient to enable the learner to carry out successfully the on-the-job tasks that require more than basic knowledge. Thus, the probability of making real errors in the business environment is high. On the other hand, if the learning environment allows the learner to experience and learn from failures within a variety of situations similar to what they would experience in the real world of their job, the probability of having similar failures in their business environment is low. This is the realm of simulations-a safe place to fail. The appearance of statistical software is one of the most important events in the process of decision making under uncertainty. Statistical software systems are used to construct examples, to understand the existing concepts, and to find new statistical properties. On the other hand, new developments in the process of decision making under uncertainty often motivate developments of new approaches and revision of the existing software systems. Statistical software systems rely on a cooperation of statisticians, and software developers. Beside the professional statistical software Online statistical computation . and the use of a scientific calculator is required for the course. A Scientific Calculator is the one, which has capability to give you, say, the result of square root of 5. Any calculator that goes beyond the 4 operations is fine for this course. These calculators allow you to perform simple calculations you need in this course, for example, enabling you to take square root, to raise e to the power of say, 0.36. and so on. These types of calculators are called general Scientific Calculators. There are also more specific and advanced calculators for mathematical computations in other areas such as Finance, Accounting, and even Statistics. The last one, for example, computes mean, variance, skewness, and kurtosis of a sample by simply entering all data one-by-one and then pressing any of the mean, variance, skewness, and kurtosis keys. Without a computer one cannot perform any realistic statistical data analysis. Students who are signing up for the course are expected to know the basics of Excel. As a starting point, you need visiting the Excel Web site created for this course. If you are challenged by or unfamiliar with Excel, you may seek tutorial help from the Academic Resource Center at 410-837-5385, E-mail. What and How to Hand-in My Computer Assignment For the computer assignment I do recommend in checking your hand computation homework, and checking some of the numerical examples from your textbook. As part of your homework assignment you don not have to hand in the printout of the computer assisted learning, however, you must include within your handing homework a paragraph entitled Computer Implementation describing your (positive or negative) experience. Interesting and Useful Sites The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia. of materials presented on this Web site is permitted for non-commercial and classroom purposes only. This site may be mirrored intact (including these notices), on any server with public access. All files are available at home. ubalt. eduntsbarshBusiness-stat for mirroring. Kindly e-mail me your comments, suggestions, and concerns. Obrigado. EOF: CopyRights 1994-2015.