7.14 Finding critical .

What critical value 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?

7.15 A one-sample test.

The one-sample statistic for testing

from a sample of observations has the value .

7.16 Another one-sample test.

The one-sample statistic for testing

from a sample of observations has the value .

7.17 A final one-sample test.

The one-sample statistic for testing

based on observations has the value .

7.18 Business bankruptcies in Canada.

Business bankruptcies in Canada are monitored by the Office of the Superintendent of Bankruptcy Canada (OSB).⁷ 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.

7.19 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 they perform the tests themselves. To keep the industry honest, the EPA does run some spot checks each year. Recently, the EPA announced that Hyundai and Kia must lower their fuel economy estimates for many of their models.⁸ Here are some city miles per gallon (mpg) values for one of the models the EPA investigated:

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

7.20 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 -value, and your conclusion.

7.21 The return-trip effect.

We often feel that the return trip from a destination takes less time than the trip to the destination even though the distance traveled is usually identical. To better understand this effect, a group of researchers ran a series of experiments.⁹ In one experiment, they surveyed 69 participants who had just returned from a day trip by bus. Each was asked to rate how long the return trip had taken, compared with the initial trip, on an 11-point scale from a lot shorter to a lot longer. The sample mean was −0.55, and the sample standard deviation was 2.16.

7.22 Health insurance costs.

The Consumer Expenditure Survey provides information on the buying habits of U.S. consumers.¹⁰ In the latest report, the average amount a husband and wife spent on health insurance was reported to be $3251 with a standard error of $89.76. Assuming a sample size of , calculate a 90% confidence interval for the average amount a husband and wife spent on health insurance.

7.23 Counts of seeds in one-pound scoops.

A leading agricultural company must maintain strict control over the size, weight, and number of seeds they package for sale to customers. An SRS of 81 one-pound scoops of seeds was collected as part of a Six Sigma quality improvement effort within the company. The number of seeds in each scoop follows.

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7.24 How many seeds on average?

Refer to the previous exercise.

7.25 Significance test for the average number of seeds.

Refer to the previous two exercises.

7.26 Investigating the Endowment Effect.

Consider an ice-cold glass of lemonade on a hot July day. What is the maximum price you’d be willing to pay for it? What is the minimum price at which you’d be willing to sell it? For most people, the maximum buying price will be less than the minimum selling price. In behavioral economics, this occurrence is called the endowment effect. People seem to add value to products, regardless of attachment, just because they own them.

As part of a series of studies, a group of researchers recruited 40 students from a graduate marketing course and asked each of them to consider a Vosges Woolloomooloo gourmet chocolate bar made with milk chocolate and coconut.¹¹ Test the null hypothesis that there is no difference between the two prices. Also construct a 95% confidence interval of the endowment effect.

7.27 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. The following is a summary of these alcohol content tests, each done on a single can of beer.

7.28 Health care costs.

The cost of health care is the subject of many studies that use statistical methods. One such study estimated that the average length of service for home health care among people aged 65 and over who use this type of service is 242 days with a standard error of 21.1 days. Assuming sample size larger than 1000, calculate a 90% confidence interval for the mean length of service for all users of home health care aged 65 and over.¹³

7.29 Plant capacity.

A leading company chemically treats its product before packaging. The company monitors the weight of product per hour that each machine treats.

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An SRS of 90 hours of production data for a particular machine is collected. The measured variable is in pounds.

7.30 Credit card fees.

A bank wonders whether omitting the annual credit card fee for customers who charge at least $5000 in a year would increase the amount charged on its credit card. The bank makes this offer to an SRS of 125 of its existing credit card customers. It then compares how much these customers charge this year with the amount that they charged last year. The mean is $685, and the standard deviation is $1128.

7.31 Supermarket shoppers.

A marketing consultant observed 40 consecutive shoppers at a supermarket. One variable of interest was how much each shopper spent in the store. Here are the data (in dollars), arranged in increasing order:

7.32 The influence of big shoppers.

Eliminate the three largest observations, and redo parts (a), (b), and (c) of the previous exercise. Do these observations have a large influence on the results?

7.33 Corn seed prices.

The U.S. Department of Agriculture (USDA) uses sample surveys to obtain important economic estimates. One USDA pilot study estimated the amount a farmer will pay per planted acre for corn seed from a sample of 20 farms. The mean price was reported as $97.59 with a standard error of $13.49. Give a 95% confidence interval for the amount a farmer will pay per planted acres for corn seed.¹⁴

7.34 Executives learn Spanish.

A company contracts with a language institute to provide instruction in Spanish for its executives who will be posted overseas. The following table gives the pretest and posttest scores on the Modern Language Association’s listening test in Spanish for 20 executives.¹⁵

7.35 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.¹⁶ 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.

7.36 Design of controls.

The design of controls and instruments has a large effect on how easily people can use them. A student project investigated this effect by asking 25 right-handed students to turn a knob (with their right hands) that moved an indicator by screw action. There were two identical instruments, one with a right-hand thread (the knob turns clockwise) and the other with a left-hand thread (the knob turns counterclockwise). The following table gives the times required (in seconds) to move the indicator a fixed distance:¹⁷

7.37 Is the difference important?

Give a 90% confidence interval for the mean time advantage of right-hand over left-hand threads in the setting of the previous exercise. Do you think that the time saved would be of practical importance if the task were performed many times—for example, by an assembly-line worker? To help answer this question, find the mean time for right-hand threads as a percent of the mean time for left-hand threads.

7.38 Confidence Interval?

As CEO, you obtain the salaries of all 31 individuals working in your marketing department. You feed these salaries into your statistical software package, and the output produced includes a confidence interval. Is this a valid confidence interval? Explain your answer.

7.39 A field trial.

An agricultural field trial compares the yield of two varieties of tomatoes for commercial use. The researchers divide in half each of eight small plots of land in different locations and plant each tomato variety on one half of each plot. After harvest, they compare the yields in pounds per plant at each location. The eight differences (Variety A − Variety B) give the following statistics: and . Is there a difference between the yields of these two varieties? Write a summary paragraph to answer this question. Make sure to include , , and the -value with degrees of freedom.