Elementary Statistics - Chapter 11 Chi Square Goodness of Fit Test

The Chi-Square Goodness-of-Fit Test is introduced as a statistical method to evaluate the fit between observed data and expected theoretical distributions. The test is exemplified by a coin flip scenario, illustrating the difference between observed and expected outcomes. The null hypothesis posits no difference between observed and expected frequencies, while the alternative hypothesis suggests a discrepancy. The test's characteristics, including its positively skewed distribution and reliance on degrees of freedom, are explained. The formula for calculating the test statistic is provided, and the process of comparing this statistic to a critical value or p-value at a given significance level (alpha) is detailed. The test is always a right-tail test, and the decision to reject or fail to reject the null hypothesis is based on whether the test statistic exceeds the critical value.

This section delves into the process of determining expected frequencies, which is essential for conducting the Chi-Square Goodness-of-Fit Test. It explains how to calculate expected frequencies when outcomes are equally likely by dividing the total number of trials by the number of categories. It also addresses how to find expected frequencies when given percentages for each category, by multiplying the total frequency by the expected probability for each category. Two examples are provided: one involving employee absences by day of the week and another regarding tax preparation methods among adults. The examples demonstrate how to sum frequencies, calculate total and expected frequencies, and interpret the results in the context of the Chi-Square Test.

The paragraph outlines the prerequisites for using the Chi-Square Goodness-of-Fit Test, emphasizing that observed frequencies must be derived from a random sample and that each expected frequency should be at least 5. It details the steps for performing the test, including finding the expected frequency, verifying the conditions for expected frequencies, stating the null and alternative hypotheses, and identifying the significance level (alpha) and degree of freedom. The process of finding the critical value from a Chi-Square distribution table is explained, and the method for calculating the test statistic using a calculator or online tool is described. The decision to reject or fail to reject the null hypothesis is based on whether the test statistic is greater than the critical value, indicating a difference between observed and expected frequencies.

This section introduces contingency tables, which are used to display observed frequencies for two categorical variables in rows and columns. The Chi-Square Test for Independence is explained as a method to determine whether there is a relationship between the two variables. The null hypothesis suggests that the variables are independent, while the alternative hypothesis posits a dependence. The process of finding expected frequencies in each cell of the table is detailed, using the formula that multiplies row and column totals and divides by the grand total. An example involving smoking habits of students related to their parents' smoking is provided, demonstrating how to calculate expected frequencies and apply the Chi-Square Test for Independence.

The paragraph presents a practical application of the Chi-Square Independence Test using data from a health club manager who wants to determine if the number of days college students exercise per week is related to gender. The null hypothesis states that the number of exercise days is independent of gender, while the alternative hypothesis suggests a dependence. The process includes stating the hypotheses, finding the critical value using the Chi-Square distribution table, and calculating the test statistic with the help of a calculator's matrix function. The test statistic is then compared to the critical value to decide whether to reject or fail to reject the null hypothesis, concluding whether there is evidence of a relationship between exercise frequency and gender.

This final paragraph summarizes the process of interpreting Chi-Square Test results in the context of hypothesis testing. It emphasizes that the test is always a right-tail test, and the decision to reject or fail to reject the null hypothesis is based on the comparison of the test statistic to the critical value. The paragraph also provides examples of how to state conclusions based on the test outcomes, such as concluding that CEOs' ages are dependent on company size when the null hypothesis is rejected. The importance of the right-tail test and the significance of the test statistic in relation to the critical value are highlighted.