7.14 What is wrong? In each of the following situations, identify what is wrong and then either explain why it is wrong or change the wording of the statement to make it true.

7.15 Finding the critical value t*. What critical value t* from Table D should be used to calculate the margin of error for a confidence interval for the mean of the population in each of the following situations?

7.16 Distribution of the t statistic. Assume a sample size of . Draw a picture of the distribution of the t statistic under the null hypothesis. Use Table D and your picture to illustrate the values of the test statistic that would lead to rejection of the null hypothesis at the 5% level for a two-sided alternative.

7.17 More on the distribution of the t statistic. Repeat the previous exercise for the two situations where the alternative is one-sided.

7.18 One-sided versus two-sided P-values. Computer software reports and for a t test of versus . Based on prior knowledge, you justified testing the alternative . What is the P-value for your significance test?

7.19 More on one-sided versus two-sided P-values. Suppose that computer software reports and for a t test of versus . Would this change your P-value for the alternative hypothesis in the previous exercise? Use a sketch of the distribution of the test statistic under the null hypothesis to illustrate and explain your answer.

7.20 A one-sample t test. The one-sample t statistic for testing

H₀: μ = 8

H_a: μ > 8

from a sample of n = 22 observations has the value t = 2.24.

7.21 Another one-sample t test. The one-sample t statistic for testing

H₀: μ = 40

H_a: μ ≠ 40

from a sample of observations has the value .

7.22 A final one-sample t test. The one-sample t statistic for testing

H₀: μ = 20

H_a: μ < 20

based on observations has the value .

7.23 Two-sided to one-sided P-value. Most software gives P-values for two-sided alternatives. Explain why you cannot always divide these P-values by 2 to obtain P-values for one-sided alternatives.

7.24 Business bankruptcies in Canada. Business bankruptcies in Canada are monitored by the Office of the Superintendent of Bankruptcy Canada (OSB).⁸ Included in each report are the assets and liabilities the company declared at the time of the bankruptcy filing. A study is based on a random sample of 75 reports from the current year. The average debt (liabilities minus assets) is $92,172 with a standard deviation of $111,538.

7.25 Fuel economy. Although the Environmental Protection Agency (EPA) establishes the tests to determine the fuel economy of new cars, it often does not perform them. Instead, the test protocols are given to the car companies, and the companies perform the tests themselves. To keep the industry honest, the EPA runs some spot checks each year. Recently, the EPA announced that Hyundai and Kia must lower their fuel economy estimates for many of their models.⁹ Here are some city miles per gallon (mpg) values for one of the models the EPA investigated:

Give a 95% confidence interval for μ, the mean city mpg for this model.

7.26 Testing the sticker information. Refer to the previous exercise. The vehicle sticker information for this model stated a city average of 30 mpg. Are these mpg values consistent with the vehicle sticker? Perform a significance test using the 0.05 significance level. Be sure to specify the hypotheses, the test statistic, the P-value, and your conclusion.

7.27 UberX driver earnings. On its blog, Uber posted a scatterplot using a sample of several thousand drivers in New York City. The plot shows each driver’s average net earnings per hour versus the number of hours worked.¹⁰ Here is a sample of earnings (dollars) for 27 drivers working 40 hours a week.

7.28 Number of friends on Facebook. To mark Facebook’s 10th birthday, Pew Research surveyed people using Facebook to see what they like and dislike about the site. The survey found that among adult Facebook users, the average number of friends is 338. This distribution takes only integer values, so it is certainly not Normal. It is also highly skewed to the right with a median of 200 friends.¹¹ Consider the following SRS of Facebook users from your large university.

7.29 Alcohol content in beer. In February 2013, two California residents filed a class-action lawsuit against Anheuser-Busch, alleging the company was watering down beers to boost profits.¹² They argued that because water was being added, the true alcohol content of the beer by volume is less than the advertised amount. For example, they alleged that Budweiser beer has an alcohol content by volume of 4.7% instead of the stated 5%. CNN, NPR, and a local St. Louis news team picked up on this suit and hired independent labs to test samples of Budweiser beer and find the alcohol content. Below is a summary of these tests each done on a single can.

7.30 Using the Internet on a computer. The Nielsen Company reported that U.S. residents aged 18 to 24 years spend an average of 32.5 hours per month using the Internet on a computer.¹³ You wonder if this it true for students at your large university because so many students use their smartphones to access the Internet. You collect an SRS of students and obtain hours with hours.

7.31 Rudeness and its effect on onlookers. Many believe that an uncivil environment has a negative effect on people. A pair of researchers performed a series of experiments to test whether witnessing rudeness and disrespect affects task performance.¹⁴ In one study, 34 participants met in small groups and witnessed the group organizer being rude to a “participant” who showed up late for the group meeting. After the exchange, each participant performed an individual brainstorming task in which he or she was asked to produce as many uses for a brick as possible in five minutes. The mean number of uses was 7.88 with a standard deviation of 2.35.

7.32 Fuel efficiency t test. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas tank was filled, and the computer was then reset.¹⁵ Here are the mpg values for a random sample of 20 of these records:

7.33 Tree diameter confidence interval. A study of 584 longleaf pine trees in the Wade Tract in Thomas County, Georgia, is described in Example 6.1 (page 342). For each tree in the tract, the researchers measured the diameter at breast height (DBH). This is the diameter of the tree at a height of 4.5 feet, and the units are centimeters (cm). Only trees with DBH greater than 1.5 cm were sampled. Here are the diameters of a random sample of 40 of these trees:

7.34 Nutritional intake among Canadian high-performance male athletes. Recall Exercise 6.74 (page 382). For one part of the study, male athletes from eight Canadian sports centers were surveyed. Their average caloric intake was 3077.0 kilocalories per day (kcal/d) with a standard deviation of 987.0. The recommended amount is 3421.7. Is there evidence that Canadian high-performance male athletes are deficient in their caloric intake?

7.35 Average number of Instagram posts. LocoWise provides social media analytics to companies and marketing agencies through a variety of online tools. One tool is the Instagram Analyzer, which allows a user to compare a profile with 2500 other Instagram profiles. Recently, it reported that the 2500 profiles it monitors averaged 2.55 posts per day, with a minimum value of 0 posts and a maximum value of 95 posts.¹⁶

7.36 Stress levels in parents of children with ADHD. In a study of parents who have children with attention-deficit/hyperactivity disorder (ADHD), parents were asked to rate their overall stress level using the Parental Stress Scale (PSS).¹⁷ This scale has 18 items that contain statements regarding both positive and negative aspects of parenthood. Respondents are asked to rate their agreement with each statement using a five-point scale (1 = strongly disagree to 5 = strongly agree). The scores are summed such that a higher score indicates greater stress. The mean rating for the 50 parents in the study was reported as 52.98 with a standard deviation of 10.34.

7.37 Are the parents feeling extreme stress? Refer to the previous exercise. The researchers considered a score greater than 45 to represent extreme stress. Is there evidence that the average stress level for the parents in this study is above this level? Perform a test of significance using and summarize your results.

7.38 Food intake and weight gain. If we increase our food intake, we generally gain weight. Nutrition scientists can calculate the amount of weight gain that would be associated with a given increase in calories. In one study, 16 nonobese adults, aged 25 to 36 years, were fed 1000 calories per day in excess of the calories needed to maintain a stable body weight. The subjects maintained this diet for eight weeks, so they consumed a total of 56,000 extra calories.¹⁸ According to theory, 3500 extra calories will translate into a weight gain of 1 pound. Therefore, we expect each of these subjects to gain 56,000/3500 = 16 pounds (lb). Here are the weights before and after the eight-week period, expressed in kilograms (kg):

7.39 Food intake and NEAT. Nonexercise activity thermogenesis (NEAT) provides a partial explanation for the results you found in the previous analysis. NEAT is energy burned by fidgeting, maintenance of posture, spontaneous muscle contraction, and other activities of daily living. In the study of the previous exercise, the 16 subjects increased their NEAT by 328 calories per day, on average, in response to the additional food intake. The standard deviation was 256.

7.40 Potential insurance fraud? Insurance adjusters are concerned about the high estimates they are receiving from Jocko’s Garage. To see if the estimates are unreasonably high, each of 10 damaged cars was taken to Jocko’s and to another garage and the estimates (in dollars) were recorded. Here are the results:

7.41 Fuel efficiency comparison t test. Refer to Exercise 7.32. In addition to the computer calculating miles per gallon, the driver also recorded the miles per gallon by dividing the miles driven by the number of gallons at fill-up. The driver wants to determine if these calculations are different.

431

7.42 Counts of picks in a one-pound bag. A guitar supply company must maintain strict oversight on the number of picks they package for sale to customers. Their current advertisement specifies between 900 and 1000 picks in every bag. An SRS of 36 one-pound bags of picks was collected as part of a quality improvement effort within the company. The number of picks in each bag is shown in the following table.

7.43 Significance test for the average number of picks. Refer to the previous exercise.

7.44 A customer satisfaction survey. Many organizations are doing surveys to determine the satisfaction of their customers. Attitudes toward various aspects of campus life were the subject of one such study conducted at Purdue University. Each item was rated on a 1 to 5 scale, with 5 being the highest rating. The average response of 1568 first-year students to “Feeling welcomed at Purdue” was 3.83 with a standard deviation of 1.10. Assuming that the respondents are an SRS, give a 90% confidence interval for the mean of all first-year students.

7.45 Comparing operators of a DXA machine. Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab, and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators.¹⁹ The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data:

7.46 Equivalence of paper and computer-based questionnaires. Computers are commonly being used to complete questionnaires because of the increased efficiency of data collection and reduction in coding errors. Studies, however, have shown that questionnaire format can influence responses, especially for items of a sensitive nature.²⁰ Consider the small study below comparing paper and computer survey formats of a self-report measure of mental health. Each participant completed both forms on adjacent days with the order determined by a flip of a coin.

7.47 Assessment of a foreign-language institute. The National Endowment for the Humanities sponsors summer institutes to improve the skills of high school teachers of foreign languages. One such institute hosted 20 French teachers for four weeks. At the beginning of the period, the teachers were given the Modern Language Association’s listening test of understanding of spoken French. After four weeks of immersion in French in and out of class, the listening test was given again. (The actual French spoken in the two tests was different, so that simply taking the first test should not improve the score on the second test.) The maximum possible score on the test is 36.²¹ Here are the data:

To analyze these data, we first subtract the pretest score from the posttest score to obtain the improvement for each teacher. These 20 differences form a single sample. They appear in the “Gain” columns. The first teacher, for example, improved from 32 to 34, so the gain is 34 − 32 = 2.