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## Chi-Square Procedures

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**Chi-Square Procedures**Chi-Square Test for Goodness of Fit, Independence of Variables, and Homogeneity of Proportions**The chi-square Goodness of Fit Test: you have only one**set of data on a single characteristic, and you want to know if it matches an expected distribution based on the laws of probability(1 variable, 1population)**In a chi-square goodness of fit test, the null hypothesis is**always Ho: The data follow a specified distribution The alternative hypothesis is always Ha: The data does not follow a specified distribution**The idea behind testing these types of claims is to compare**actual counts to the counts we would expect if the null hypothesis were true. If a significant difference between the actual counts and expected counts exists, we would take this as evidence against the null hypothesis.**The method for obtaining the expected counts requires that**we determine the number of observations within each cell under the assumption the null hypothesis is true.**Test Statistic for the Test of Goodness of Fit**Let Oi represent the observed number of counts in the ith cell, Ei represent the expected number of counts in the ith cell. Then, approximately follows the chi-square distribution with(# of cells– 1) degrees of freedom in the contingency table**The Chi-Square Test for Goodness of Fit**If a claim is made regarding the data following a certain distribution, we can use the following steps to test the claim provided 1. the data is randomly selected**The Chi-Square Test for Goodness of Fit**If a claim is made regarding the data following a certain distribution, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1.**The Chi-Square Test for Goodness of Fit**If a claim is made regarding the data following a certain distribution, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1. 3. 80% of the expected cell counts are greater than or equal to 5.**EXAMPLE Testing for Goodness of Fit**In consumer marketing, a common problem that any marketing manager faces is the selection of appropriate colors for package design. Assume that a marketing manager wishes to compare five different colors of package design. He is interested in knowing if there is a preference among the five colors so that it can be introduced in the market. A random sample of 400 consumers reveals the following. Do the consumer preferences for package colors show any significant difference?**Step 1. A claim is made regarding the data fit to a**certain distribution. Ho: Ha:**Step 1. A claim is made regarding the data fit to a**certain distribution. Ho: the number of customers who prefer each color are the same. Ha: the number of customers who prefer each color are not the same.**Step 2: Calculate the expected frequencies (counts) for each**cell in the contingency table.**Step 2: Calculate the expected frequencies (counts) for each**cell in the contingency table. Observed Counts Expected Counts**Step 3: Verify the requirements for the chi-square test for**goodness of fit are satisfied. (1) data is randomly selected (2) all expected frequencies are greater than or equal to 1 (3) 80% of the expected cell counts are greater than or equal to 5. Step 4: Select a proper level of significance **Step 5: Compute the test statistic and P-value**P-value = cdf(min,max,df)**Step 5: Compute the test statistic and P-value**P-value = 0.0224 11.4**If P-value < , reject null hypothesis**11.4>9.49 and 0.0224<0.05. Therefore I would reject the null hypothesis. The data is statistically significant and I am led to believe that there is a difference in preference of package color**The chi-square independence test: you have two**characteristics of a population, and you want to see if there is any association between the characteristics(2 variables, 1 population)**In a chi-square independence test, the null hypothesis is**always Ho: the variables are independent The alternative hypothesis is always Ha: the variables are dependent**The idea behind testing these types of claims is to compare**actual counts to the counts we would expect if the null hypothesis were true (if the variables are independent). If a significant difference between the actual counts and expected counts exists, we would take this as evidence against the null hypothesis.**The method for obtaining the expected counts requires that**we determine the number of observations within each cell under the assumption the null hypothesis is true.**Expected Frequencies in a Chi-Square Independence Test**To find the expected frequencies in a cell when performing a chi-square independence test, multiply the row total of the row containing the cell by the column total of the column containing the cell and divide this result by the table total. That is**Test Statistic for the Test of Independence**Let Oi represent the observed number of counts in the ith cell, Ei represent the expected number of counts in the ith cell. Then, approximately follows the chi-square distribution with(r – 1)(c – 1) degrees of freedom where r is the number of rows and c is the number of columns in the contingency table**The Chi-Square Test for Independence**If a claim is made regarding the association between (or independence of) two variables in a contingency table, we can use the following steps to test the claim provided 1. the data is randomly selected**The Chi-Square Test for Independence**If a claim is made regarding the association between (or independence of) two variables in a contingency table, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1.**The Chi-Square Test for Independence**If a claim is made regarding the association between (or independence of) two variables in a contingency table, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1. 3. 80% of the expected cell counts are greater than or equal to 5.**Step 1. A claim is made regarding the independence of the**data. Ho: Ha:**Step 1. A claim is made regarding the independence of the**data. Ho: there is not association between gender of lifestyle choice, the variables are independent Ha: there is an association between gender of lifestyle choice, the variables are dependent**Step 2: Calculate the expected frequencies (counts) for each**cell in the contingency table.**Step 2: Calculate the expected frequencies (counts) for each**cell in the contingency table. Observed Counts Expected Counts**Step 3: Verify the requirements for the chi-square test for**independence are satisfied. (1) data is randomly selected (2) all expected frequencies are greater than or equal to 1 (3) 80% of the expected cell counts are greater than or equal to 5. Step 4: Select a proper level of significance **Step 5: Compute the test statistic and P-Value**P-value = cdf(min,max,df)**Step 5: Compute the test statistic and P-Value**36.84 P = 0.00000001**If P-value < , reject null hypothesis**36.84>5.99 and 0.00000001<0.05. Therefore I would reject the null hypothesis. The data is statistically significant and I am led to believe that there is an association between gender and lifestyle choice and that these variables are dependent**In a chi-square test for homogeneity:you take samples from**different populations, and you want to test to see if the proportions in various categories is the same for each population(1 variable, multiple populations)**In a chi-square homogeneity test, the null hypothesis is**always Ho: populations have the same proportion of individuals with some characteristic. The alternative hypothesis is always Ha: populations have different proportion of individuals with some characteristic.**The idea behind testing these types of claims is to compare**actual counts to the counts we would expect if the null hypothesis were true (proportions are equal). If a significant difference between the actual counts and expected counts exists, we would take this as evidence against the null hypothesis.**The method for obtaining the expected counts requires that**we determine the number of observations within each cell under the assumption the null hypothesis is true.**Expected Frequencies in a Chi-Square Homogeneity Test**To find the expected frequencies in a cell when performing a chi-square independence test, multiply the row total of the row containing the cell by the column total of the column containing the cell and divide this result by the table total. That is**Test Statistic for the Test of Homogeneity**Let Oi represent the observed number of counts in the ith cell, Ei represent the expected number of counts in the ith cell. Then, approximately follows the chi-square distribution with(r – 1)(c – 1) degrees of freedom where r is the number of rows and c is the number of columns in the contingency table**The Chi-Square Test forHomogeneity**If a claim is made regarding that different populations have the same proportion of individuals with some characteristic, we can use the following steps to test the claim provided 1. the data is randomly selected**The Chi-Square Test for Homogeneity**If a claim is made regarding that different populations have the same proportion of individuals with some characteristic, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1.**The Chi-Square Test for Homogeneity**If a claim is made regarding that different populations have the same proportion of individuals with some characteristic, we can use the following steps to test the claim provided 1. the data is randomly selected 2. all expected frequencies are greater than or equal to 1. 3. 80% of the expected cell counts are greater than or equal to 5.**EXAMPLE A Test of Homogeneity of Proportions**The following question was asked of a random sample of individuals in 1992, 1998, and 2001: “Would you tell me if you feel being a teacher is an occupation of very great prestige?” The results of the survey are presented below:**Step 1. A claim is made regarding the homogeneity of the**data. Ho: Ha: