Example 10.1a Experimenting with a New Pizza Style at the Pepperoni Pizza Restaurant Hypothesis Tests for a Population Mean

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Objective To use a one-sample t test to see whether consumers prefer the new style pizza to the old style.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Background Information n Recall that the manager of the Pepperoni Pizza Restaurant is running an experiment to test the hypotheses of H 0 : mu  0 versus H a : mu> 0, where mu is the mean rating in the entire customer population. n Here, each customer rates the difference between an old-style pizza and a new-style pizza on a -10 to +10 scale, where negative ratings favor the old-style pizza and positive ratings favor the new-style pizza.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | PIZZA.XLS n The ratings of 40 randomly selected customers and several summary statistics appear in this file and in the following table.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Summary Statistics n From the summary statistics, we see that the sample mean is 2.10 and the sample standard deviation is n The positive sample mean provides some evidence in favor of the alternative hypothesis, but given the rather large standard deviation and the boxplot of ratings shown on the next slide does it provide enough evidence to reject H 0 ?

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Summary Statistics -- continued

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Running the Test n To run the test, we calculate the test statistic, using the borderline null hypothesis value mu 0 = 0, and report how much probability is beyond it in the right tail of the appropriate t distribution. n We use the right tail because the alternative is one- tailed of the “greater than” variety. n The test statistic is

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Running the Test -- continued n The probability beyond this value in the right tail of the t distribution with n-1 = 39 degrees of freedom is approximately 0.004, which can be found in Excel with the function TDIST(2.816,39,1). n The probability, 0.004, is the p-value for the test. It indicates that these sample results would be very unlikely if the null hypothesis is true. n The manager has two choices: he can conclude that the null hypothesis is true or he can conclude that the alternative hypothesis is true - and presumably switch to the new-style pizza. The second choice appears to be more reasonable.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Using StatPro n Another way to interpret the results is in terms of traditional significance levels but the p-value is the preferred method. n The StatPro One-Sample procedure can be used to perform this analysis easily. To use it select the StatPro/Statistical Inference/One-Sample Analysis menu item, and choose the Rating variable as the variable to analyze. n Then fill in the dialog boxes as shown here.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | One-Sample Dialog Box

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Hypothesis Test Dialog Box

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | The Results n Most of this output should be familiar; it mirrors the previous calculations. n The results are significant at the 1% level.

| 10.2 | 10.3 | 10.4 | 10.5 | 10.6 | 10.7 | 10.8 | Conclusion n Should the manager switch to the new-style pizza on the basis of these sample results? n We would probably recommend “yes”. There is no indication that the new-style pizza costs any more to make than the old-style pizza, and the sample evidence is fairly convincing that customers, on average, will prefer the new-style pizza. n Therefore, unless there are reasons for not switching (for example, costs) then we recommend the switch.