Question

For each of the following scenarios, identify the populations, the counts, and the sample sizes; compute the two proportions and find their difference.

a) A study of tipping behaviors examined the relationship between the color of the shirt worn by the server and whether or not the customer left a tip.¹⁹ There were 418 male customers in the study; 40 of the 69 who were served by a server wearing a red shirt left a tip. Of the 349 who were served by a server wearing a different colored shirt, 130 left a tip.

b) A sample of 40 runners will be used to compare two new routines for stretching. The runners will be randomly assigned to one of the routines which they will follow for two weeks. Satisfaction with the routines will be measured using a questionnaire at the end of the two-week period. For the first routine, nine runners said that they were satisfied or very satisfied. For the second routine, six runners said that they were satisfied or very satisfied.

c) For each scenario, find the large-sample 95% confidence interval for the difference in proportions and use the scenario to explain the meaning of the confidence interval.

d) For each scenario, perform the large-sample significance test and use the scenario to explain the meaning of the significance test.

need answer for c) and d).

Answer 1

For scenario a)

Here, 40 of the 69 who were served by a server wearing a red shirt left a tip. Of the 349 who were served by a server wearing a different colored shirt, 130 left a tip.

Therefore:

The sample sizes are:

Two proportions and their difference are:

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c) For 95% confidence level, critical value of z is .

The large-sample 95% confidence interval for the difference in proportions is:

We are 95% confident that the difference in proportion of male customers left a tip who were served by a server wearing a red shirt and who were served by a server wearing a different colored shirt lies between 0.081 and 0.335. Since 0 does not lie in the confidence interval, we can conclude that higher proportion of male customers left a tip who were served by a server wearing a red shirt than those who were served by a server wearing a different colored shirt.

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d) The hypotheses are:

The pooled proportion is:

The test statistic is:

The p-value is:

Since p-value is less than 0.05, reject the null hypothesis. We can conclude that there is significant difference in proportion of male customers left a tip who were served by a server wearing a red shirt and who were served by a server wearing a different colored shirt.