SECTION 8.2 Exercises

For Exercises 8.49 to 8.51, see page 437; for 8.52 and 8.53, see pages 439–440; for 8.54 to 8.56, see page 440; for 8.57 to 8.58, see page 444; for 8.59, see page 445; and for 8.60, see page 447.

Question 8.61

8.61 To tip or not to tip

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.15 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.

  1. What is the explanatory variable for this setting? Explain your answer.
  2. What is the response variable for this setting? Explain your answer.
  3. What are the parameters for this study? Explain your answer.

8.61

(a) The explanatory variable is the color of the shirt; it is used to explain whether or not a tip was given. (b) The response variable is whether a tip was given or not; it is the interest of the study. (c) is the true proportion of male customers who would leave a tip for a red-shirt server. is the true proportion of male customers who would leave a tip for a different-colored-shirt server.

Question 8.62

8.62 Confidence interval for tipping

Refer to the previous exercise.

  1. Find the proportion of tippers for the red-shirted servers and the proportion of tippers for the servers with other colored shirts.
  2. Find a 95% confidence interval for the difference in proportions.
  3. Write a short paragraph summarizing your results.

Question 8.63

8.63 Significance test for tipping

Refer to the previous two exercises.

  1. Give a null hypothesis for this setting in terms of the parameters. Explain the meaning of the null hypothesis in simple terms.
  2. Give an alternative hypothesis for this setting in terms of the parameters. Explain the meaning of the alternative hypothesis in simple terms. Give a reason for your choice of this particular alternative hypothesis.
  3. Are the conditions satisfied for the use of the significance test based on the Normal distribution? Explain your answer.

8.63

(a) . For male customers, the percent who tip a red-shirted server is the same as the percent who tip a different-colored-shirt server. (b) Answers will vary. . For male customers, the percent who tip a red-shirted server is higher than the percent who tip a different-coloredshirt server. Because the color red was singled out, it is logical that we might expect it to have higher percent of tips. (Note: is also acceptable with a suitable argument.) (c) Yes, we have at least 10 successes and failures in each sample.

Question 8.64

8.64 Significance test details for tipping

Refer to the previous exercise.

  1. Find the test statistic.
  2. What is the distribution of the test statistic if the null hypothesis is true?
  3. Find the -value.
  4. Use a sketch of a Normal distribution to explain the interpretation of the -value that you found in part (c).

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  5. Write a brief summary of the results of your significance test. Include enough details so that someone reading your summary could reproduce all your results.

Question 8.65

8.65 Draw a picture

Suppose that there are two binomial populations. For the first, the true proportion of successes is 0.3; for the second, it is 0.4. Consider taking independent samples from these populations, 40 from the first and 50 from the second.

  1. Find the mean and the standard deviation of the distribution of
  2. This distribution is approximately Normal. Sketch this Normal distribution, and mark the location of the mean.
  3. Find a value for which the probability is 0.95 that the difference in sample proportions is within . Mark these values on your sketch.

8.65

(a) . (c) 0.1965.

Question 8.66

8.66 What's wrong

For each of the following, explain what is wrong and why.

  1. A 95% confidence interval for the difference in two proportions includes errors due to bias.
  2. A t statistic is used to test the null hypothesis that
  3. If two sample counts are equal, then the sample proportions are equal.

Question 8.67

8.67 College student summer employment.

Suppose that 83% of college men and 80% of college women were employed last summer. A sample survey interviews SRSs of 300 college men and 300 college women. The two samples are independent.

  1. What is the approximate distribution of the proportion of women who worked last summer? What is the approximate distribution of the proportion of men who worked?
  2. The survey wants to compare men and women. What is the approximate distribution of the difference in the proportions who worked, ?

8.67

(a) . (b) .

Question 8.68

8.68 A corporate liability trial

A major court case on liability for contamination of groundwater took place in the town of Woburn, Massachusetts. A town well in Woburn was contaminated by industrial chemicals. During the period that residents drank water from this well, there were 16 birth defects among 414 births. In years when the contaminated well was shut off and water was supplied from other wells, there were three birth defects among 228 births. The plaintiffs suing the firms responsible for the contamination claimed that these data show that the rate of birth defects was higher when the contaminated well was in use.16 How statistically significant is the evidence? Be sure to state what assumptions your analysis requires and to what extent these assumptions seem reasonable in this case.

Question 8.69

8.69 Natural versus artificial Christmas trees.

CASE 8.2 In the Christmas tree survey introduced in Case 8.2 (page 428), respondents who had a tree during the holiday season were asked whether the tree was natural or artificial. Respondents were also asked if they lived in an urban area or in a rural area. Of the 421 households displaying a Christmas tree, 160 lived in rural areas and 261 were urban residents. The tree growers want to know if there is a difference in preference for natural trees versus artificial trees between urban and rural households. Here are the data:

Population (natural)
1 (rural) 160 64
2 (urban) 261 89
  1. Give the null and alternative hypotheses that are appropriate for this problem, assuming we have no prior information suggesting that one population would have a higher preference than the other.
  2. Test the null hypothesis. Give the test statistic and the -value, and summarize the results.
  3. Give a 90% confidence interval for the difference in proportions.

8.69

(a) . (b) . The data do not show a difference in preference for natural trees versus artificial trees between urban and rural households. (c) (−0.021, 0.139).

Question 8.70

8.70 Summer employment of college students

A university financial aid office polled an SRS of undergraduate students to study their summer employment. Not all students were employed the previous summer. Here are the results for men and women:

Men Women
Employed 622 533
Not employed 58 82
Total 680 615
  1. Is there evidence that the proportion of male students employed during the summer differs from the proportion of female students who were employed? State and , compute the test statistic, and give the -value.
  2. Give a 95% confidence interval for the difference between the proportions of male and female students who were employed during the summer. Does the difference seem practically important to you?

Question 8.71

8.71 Effect of the sample size

Refer to the previous exercise. Similar results from a smaller number of students may not have the same statistical significance. Specifically, suppose that 124 of 136 men surveyed were employed and 106 of 122 women surveyed were employed. The sample proportions are essentially the same as in the earlier exercise.

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  1. Compute the statistic for these data, and report the -value. What do you conclude?
  2. Compare the results of this significance test with your results in Exercise 8.70. What do you observe about the effect of the sample size on the results of these significance tests?

8.71

(a) . The data show no significant difference between the proportions of male and female students employed during the summer. (b) The smaller sample sizes are not big enough to detect the practical difference we observed in the previous exercise.

Question 8.72

8.72 Find the power

Consider testing the null hypothesis that two proportions are equal versus the two-sided alternative with , 80% power, and equal sample sizes in the two groups.

  1. For each of the following situations, find the required sample size: (i) and , (ii) and , (iii) and , (iv) and , (v) and , (vi) and , (vii) and , and (viii) and .
  2. Write a short summary describing your results.