CHAPTER 8 Review Exercises

Question 8.73

8.73 The Internet of Things

The Internet of Things (IoT) refers to connecting computers, phones, and many other types of devices so that they can communicate and interact with each other.17 A Pew Internet study asked a panel of 1,606 experts whether they thought that the IoT would have “widespread and beneficial effects on the everyday lives of the public by 2025.” Eighty-three percent of the panel gave a positive response.18

  1. How many of the experts responded Yes to the question? Show your work.
  2. Describe the population proportion for this setting.
  3. What is the sample proportion? Show your work.
  4. What is the standard error for the sample proportion? Show your work.
  5. What is the margin of error for 95% confidence? Show your work.
  6. Give the 95% confidence interval for the population proportion.

8.73

(a) or 1333. (b) is the true proportion of all experts who would give a positive response to the question. (c) ; is given in the problem. (d) (e) . (f) (0.812, 0.848).

Question 8.74

8.74 A new Pew study of Internet of Things

Refer to the previous exercise. Suppose Pew would like to do a new study next year to see if expert opinion has changed since the original study was performed. Assume that a new panel of 1606 experts would be asked the same question.

  1. Using 95% confidence, compute the margin of error for the difference in proportions between the two studies for each of the following possible values of the sample proportion for the new study: (i) 0.77, (ii) 0.79, (iii) 0.81, (iv) 0.83, (v) 0.85, (vi) 0.87, and (vii) 0.89.
  2. Summarize your results with a graph.
  3. Write a short summary describing what you have found in this exercise.

Question 8.75

8.75 Find the power

Refer to the previous exercise. Consider performing a significance test to compare the population proportions for the two studies. Use and a two-sided alternative.

  1. Find the power of the significance test for each of the following population proportions in the new study: (i) 0.77, (ii) 0.79, (iii) 0.81, (iv) 0.83, (v) 0.85, (vi) 0.87, and (vii) 0.89.
  2. Display your results with a graph.
  3. Write a short summary describing what you have found in this exercise.

8.75

(a) (i) 0.99. (ii) 0.82. (iii) 0.31. (iv) 0.03. (v) 0.34. (vi) 0.89. (vii) 1. (c) As the difference between the two proportions gets smaller, the power decreases; if the two proportions are quite different, then the power is quite large.

Question 8.76

8.76 Worker absences and the bottom line

A survey of 1234 companies found that 36% of them did not measure how worker absences affect their company's bottom line.19

  1. How many of the companies responded that they do measure how worker absences affect their company's bottom line? Show your work.
  2. Describe the population proportion for this setting.
  3. What is the sample proportion? Show your work.
  4. What is the standard error for the sample proportion? Show your work.
  5. What is the margin of error for 95% confidence? Show your work.
  6. Give the 95% confidence interval for the population proportion.

Question 8.77

8.77 The new worker absence study

Refer to the previous exercise. Suppose you would like to do a new study next year to see if there has been a change in the percent of companies that do not measure how worker absences affect their company's bottom line. Assume that a new sample of 1234 companies will be used for the new study.

  1. Compute the 95% margin of error of the difference in proportions between the two studies for each of the following possible values of the sample proportion for the new study: (i) 0.36, (ii) 0.41, (iii) 0.46, and (iv) 0.51.
  2. Summarize your results with a graph.

    452

  3. Write a short summary describing what you have found in this exercise.

8.77

(a) (i) 0.0379. (ii) 0.0383. (iii) 0.0386. (iv) 0.0387. (c) As the second proportion increases (moves closer to 0.5), the margin of error of the difference in proportions increases somewhat, but note the change is not drastic due to the large sample size.

Question 8.78

8.78 Find the power

Refer to the previous exercise. Consider performing a significance test to compare the population proportions for the two studies. Use and a one-sided alternative.

  1. Find the power of the significance test for each of the following population proportions in the new study: (i) 0.36, (ii) 0.41, (iii) 0.46, and (iv) 0.51.
  2. Display your results with a graph.
  3. Write a short summary describing what you have found in this exercise.

Question 8.79

8.79 Worker absences and the bottom line

Refer to Exercises 8.76 through 8.78. Suppose that the companies participating in the new studies are the same as the companies in the original study. Would your answers to any of the parts of Exercises 8.77 and 8.78 change? Explain your answer.

8.79

The results in the previous exercises assume independent samples; if the same companies were used, this would not be the case and the results from comparing the two studies would not be valid.

Question 8.80

8.80 Effect of the Fox News app

A survey that sampled smartphone users quarterly compared the proportions of smartphone users who visited the Fox News website before and after the introduction of a Fox News app. A report of the survey stated that 17.6% of smartphone users visited the Fox News website before the introduction of the app versus 18.5% of users after the app was introduced.20 Assume that the sample sizes were 5600 for each condition.

  1. What is the explanatory variable for this study?
  2. What is the response variable for this study?
  3. Give a 95% confidence interval for the difference in the proportions.

Question 8.81

8.81 A significance test for the Fox News app

Refer to the previous exercise.

  1. State an appropriate null hypothesis for this setting.
  2. Give an alternative hypothesis for this setting. Explain the meaning of the alternative hypothesis in simple terms, and explain why you chose this particular alternative hypothesis.
  3. Are the conditions satisfied for the use of the significance test based on the Normal distribution?

8.81

(a) . (b) . The proportion of smartphone users who visit the Fox News website before the Fox News app was introduced is less than the proportion of smartphone users who visit the Fox News website after the Fox News app was introduced. It is likely that the app helps bring traffic to the Fox News website, hence the one-sided alternative. (Note: is also acceptable with a suitable argument.) (c) Assuming the samples were random and the samples were independent (that is, not the same group), then the conditions for the Normal distribution are met because we have at least 10 successes and failures in each sample.

Question 8.82

8.82 Perform the significance test for the Fox News app

Refer to the previous exercise.

  1. What is 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 the Normal distribution to explain the interpretation or the -value that you calculated in part (c).
  5. Write a brief summary of the results of your significance test. Include enough detail so that someone reading your summary could reproduce all your results.

Question 8.83

8.83 Power for a similar significance test

Refer to Exercises 8.80, 8.81, and 8.82. Suppose you were planning a similar study for a different app. Assume that the population proportions are the same as the sample proportion in the Fox News study. The numbers of smartphone users will be the same for the before and after groups. Assume 80% power with a test using . Find the number of users needed for each group.

8.83

22,580 users for each group.

Question 8.84

8.84 What would the margin of error be?

Refer to the previous exercise. Using the sample sizes for the two groups that you found there, what would you expect the 95% margin of error to be for the estimated difference between the two proportions? For your calculations, assume that the sample proportions would be the same as given for the original setting in Exercise 8.80.

Question 8.85

8.85 The parrot effect: how to increase your tips.

An experiment examined the relationship between tips and server behavior in a restaurant.21 In one condition, the server repeated the customer's order word for word, while in the other condition, the orders were not repeated. Tips were received in 47 of the 60 trials under the repeat condition and in 31 of the 60 trials under the no-repeat condition.

  1. Find the sample proportions, and compute a 95% confidence interval for the difference in population proportions.
  2. Use a significance test to compare the two conditions. Summarize the results.

8.85

(a) .
(b) (two-sided), so reject the null hypothesis. The data show a significant difference between the proportion of customers who tip when the order was repeated and the proportion of customers who tip when the order is not repeated. The confidence interval showed an increase of between 10.3% and 43.1% in tips when the order is repeated.

Question 8.86

8.86 The parrot effect: how to increase your tips, continued

Refer to the previous exercise.

  1. The study was performed in a restaurant in The Netherlands. Two waitresses performed the tasks. How do these facts relate to the type of conclusions that can be drawn from this study? Do you think that the parrot effect would apply in other countries?
  2. Design a study to test the parrot effect in a setting that is familiar to you. Be sure to include complete details about how the study will be conducted and how you will analyze the results.

Question 8.87

8.87 Does the new process give a better product?

Twelve percent of the products produced by an industrial process over the past several months fail to conform to the specifications. The company modifies the process in an attempt to reduce the rate of nonconformities. In a trial run, the modified process produces 16 nonconforming items out of a total of 300 produced. Do these results demonstrate that the modification is effective? Support your conclusion with a clear statement of your assumptions and the results of your statistical calculations.

453

8.87

We are assuming items are independent and the 300 produced represent a random sample for the modified process. . The data show significant evidence that the proportion of nonconforming items in the modified process is less than 12%; the modification was effective.

Question 8.88

8.88 How much is the improvement?

In the setting of the previous exercise, give a 95% confidence interval for the proportion of nonconforming items for the modified process. Then, taking to be the old proportion and the proportion for the modified process, give a 95% confidence interval for .

Question 8.89

8.89 Choosing sample sizes

For a single proportion, the margin of error of a confidence interval is largest for any given sample size and confidence level C when This led us to use for planning purposes. A similar result is true for the two-sample problem. The margin of error of the confidence interval for the difference between two proportions is largest when Use these conservative values in the following calculations, and assume that the sample sizes and have the common value . Calculate the margins of error of the 95% confidence intervals for the difference in two proportions for the following choices of . Present the results in a table and with a graph. Summarize your conclusions.

8.89

For 40: 0.2191. For 80: 0.1550. For 160: 0.1096. For 320: 0.0775. For 640: 0.0548. As the sample size increases, the margin of error decreases.

Question 8.90

8.90 Choosing sample sizes, continued

As the previous exercise noted, using the guessed value 0.5 for both and gives a conservative margin of error in confidence intervals for the difference between two population proportions. You are planning a survey and will calculate a 95% confidence interval for the difference in two proportions when the data are collected. You would like the margin of error of the interval to be less than or equal to 0.05. You will use the same sample size for both populations.

  1. How large a value of is needed?
  2. Give a general formula for in terms of the desired margin of error and the critical value .

Question 8.91

8.91 Unequal sample sizes

You are planning a survey in which a 95% confidence interval for the difference between two proportions will present the results. You will use the conservative guessed value 0.5 for and in your planning. You would like the margin of error of the confidence interval to be less than or equal to 0.10. It is very difficult to sample from the first population, so that it will be impossible for you to obtain more than 25 observations from this population. Taking , can you find a value of that will guarantee the desired margin of error? If so, report the value; if not, explain why not.

8.91

It is not possible. The formula for the margin of error is given by

Plugging in 0.1 for gives , which is not possible.

Question 8.92

8.92 Students change their majors

In a random sample of 890 students from a large public university, it was found that 404 of the students changed majors during their college years.

  1. Give a 99% confidence interval for the proportion of students at this university who change majors.
  2. Express your results from part (a) in terms of the percent of students who change majors.
  3. University officials are more interested in the number of students who change majors than in the proportion. The university has 30,000 undergraduate students. Convert your confidence interval in part (a) to a confidence interval for the number of students who change majors during their college years.

Question 8.93

8.93 Statistics and the law

Casteneda v. Partida is an important court case in which statistical methods were used as part of a legal argument. When reviewing this case, the Supreme Court used the phrase “two or three standard deviations” as a criterion for statistical significance. This Supreme Court review has served as the basis for many subsequent applications of statistical methods in legal settings. (The two or three standard deviations referred to by the Court are values of the statistic and correspond to -values of approximately 0.05 and 0.0026.) In Casteneda the plaintiffs alleged that the method for selecting juries in a county in Texas was biased against Mexican Americans.22 For the period of time at issue, there were 181,535 persons eligible for jury duty, of whom 143,611 were Mexican Americans. Of the 870 people selected for jury duty, 339 were Mexican Americans.

  1. What proportion of eligible jurors were Mexican Americans? Let this value be .
  2. Let be the probability that a randomly selected juror is a Mexican American. The null hypothesis to be tested is . Find the value of for this problem, compute the statistic, and find the -value. What do you conclude? (A finding of statistical significance in this circumstance does not constitute proof of discrimination. It can be used, however, to establish a prima facie case. The burden of proof then shifts to the defense.)
  3. We can reformulate this exercise as a two-sample problem. Here, we wish to compare the proportion of Mexican Americans among those selected as jurors with the proportion of Mexican Americans among those not selected as jurors. Let be the probability that a randomly selected juror is a Mexican American, and let be the probability that a randomly selected nonjuror is a Mexican American. Find the statistic and -value. How do your answers compare with your results in part (b)?

    454

8.93

(a) . (b) . The data show evidence that the proportion of jurors selected who are Mexican American is significantly less than 79.1% (the percent of Mexican Americans in the population). (c) . The answer is nearly identical to part (b).

Question 8.94

8.94 The future of gamification as a marketing tool.

Gamification is an interactive design that includes rewards such as points, payments, and gifts. A Pew survey of 1021 technology stakeholders and critics was conducted to predict the future of gamification. A report on the survey said that 42% of those surveyed thought that there would be no major increases in gamification by 2020. On the other hand, 53% said that they believed that there would be significant advances in the adoption and use of gamification by 2020.23 Analyze these data using the methods that you learned in this chapter, and write a short report summarizing your work.

Question 8.95

8.95 Where do you get your news?

A report produced by the Pew Research Center's Project for Excellence in Journalism summarized the results of a survey on how people get their news. Of the 2342 people in the survey who own a desktop or laptop, 1639 reported that they get their news from the desktop or laptop.24

  1. Identify the sample size and the count.
  2. Find the sample proportion and its standard error.
  3. Find and interpret the 95% confidence interval for the population proportion.
  4. Are the guidelines for use of the large-sample confidence interval satisfied? Explain your answer.

8.95

(a) . (b) . (c) (0.681, 0.718). With 95% confidence, the proportion of people who get their news from the desktop or laptop is between 68.1% and 71.8%. (d) Yes, we have at least 10 successes and failures in the sample.

Question 8.96

8.96 Should you bet on Punxsutawney Phil

There is a gathering every year on February 2 at Gobbler's Knob in Punxsutawney, Pennsylvania. A groundhog, always named Phil, is the center of attraction. If Phil sees his shadow when he emerges from his burrow, tradition says that there will be six more weeks of winter. If he does not see his shadow, spring has arrived. How well has Phil predicted the arrival of spring for the past several years? The National Oceanic and Atmospheric Administration has collected data for the 25 years from 1988 to 2012. For each year, whether or not Phil saw his shadow is recorded. This is compared with the February temperature for that year, classified as above or below normal. For 18 of the 25 years, Phil saw his shadow, and for six of these years, the temperature was below normal. For the years when Phil did not see his shadow, two of these years had temperatures below normal.25 Analyze the data, and write a report on how well Phil predicts whether or not winter is over.