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The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 10: Q. 29 (page 626)

A sample survey interviews SRSs of $500$ female college students and $550$ male college students. Each student is asked whether he or she worked for pay last summer. In all, $410$ of the women and $484$ of the men say 鈥淵es.鈥� Take $p_{M}$ and localid="1650271716184" $p_{F}$ be the proportions of all college males and females who worked last summer. We conjectured before seeing the data that men are more likely to work. The hypotheses to be tested are
(a) $H_{0} : p_{M} - p_{f} = 0$ versus $H_{a}; p_{M} - p_{F} 鈮� 0$
(b) $H_{0} : p_{M} - p_{F} = 0$ versus $H_{a} : p_{M} - p_{F} > 0$ .
(c) $H_{0} : p_{M} - p_{F} = 0$ versus $H_{a} : p_{M} - p_{F} < 0$
(d) $H_{0} : p_{M} - p_{F} > 0$ versus $H_{a} : p_{M} - p_{F} = 0$
(e) $H_{0} : p_{M} - p_{F} 鈮� 0$ versus $H_{a} : p_{M} - p_{F} = 0$

Short Answer

Expert verified

The tested hypothesis is option (b) $H_{0} : p_{M} - p_{F} = 0, H_{a} : p_{M} - p_{F} > 0 .$

Step by step solution

01

Given information

Male proportions in the college $= p_{M} = 484$

Female proportion in the college $= p_{F}$ $= 410$

To find the correct hypothesis.

02

Explanation

The equal sign is always used in the null hypothesis

$H_{0} : p_{M} - p_{F} = 0$

The alternative hypothesis contradicts the null hypothesis (according to the claim)

$H_{a} : p_{M} - p_{F} > 0$

Because of $p_{M} - p_{F} > 0$

Then, $p_{M} > p_{F}$

So it is option (b) $H_{0} : p_{M} - p_{F} = 0, H_{a} : p_{M} - p_{F} > 0 .$

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Most popular questions from this chapter

A fast-food restaurant uses an automated filling machine to pour its soft drinks. The machine has different settings for small, medium, and large drink cups. According to the machine鈥檚 manufacturer, when the large setting is chosen, the amount of liquid dispensed by the machine follows a Normal distribution with mean $27$ ounces and standard deviation $0.8$ ounces. When the medium setting is chosen, the amount of liquid dispensed follows a Normal distribution with mean $17$ ounces and standard deviation $0.5$ ounces. To test the manufacturer鈥檚 claim, the restaurant manager measures the amount of liquid in a random sample of $25$ cups filled with the medium setting and a separate random sample of $20$ cups filled with the large setting. Let $x_{1} - x_{2}$ be the difference in the sample mean amount of liquid under the two settings (large 鈥� medium). What is the shape of the sampling distribution of ${\overset{炉}{x}}_{1} - {\overset{炉}{x}}_{2}$ . Why?

46. Happy customers As the Hispanic population in the United States has grown, businesses have tried to understand what Hispanics like. One study inter-viewed a random sample of customers leaving a bank. Customers were classified as Hispanic if they preferred to be interviewed in Spanish or as Anglo if they preferred English. Each customer rated the importance
of several aspects of bank service on a 10-point scale. Here are summary results for the importance of 鈥渞eliability鈥� (the accuracy of account records and so on):

(a) The distribution of reliability ratings in each group is not Normal. A graph of the data reveals no outliers. The use of two-sample t procedures is
justified. Why?
(b) Construct and interpret a $95 %$ confidence interval for the difference between the mean ratings of the importance of reliability for Anglo and
Hispanic bank customers.
(c) Interpret the $95 %$ confidence level in the context of this study.

Expensive ads Consumers who think a product鈥檚 advertising is expensive often also think the product must be of high quality. Can other information undermine this effect? To find out, marketing researchers did an experiment. The subjects were 90 women from the clerical and administrative staff of a large organization. All subjects read an ad that described a fictional line of food products called 鈥淔ive Chiefs.鈥� The ad also described the major TV commercials that would soon be shown, an unusual expense for this type of product. The $45$ women who were randomly assigned to the control group read nothing else. The $45$ in the 鈥渦ndermine group鈥� also read a news story headlined 鈥淣o Link between Advertising Spending and New Product Quality.鈥� All the subjects then rated the quality of Five Chefs products on a seven-point scale. The study report said, 鈥淭he mean quality ratings were significantly lower in the undermine treatment (xA 4.56) than in the control treatment $({\overset{炉}{x}}_{C} = 5.05; t = 2.64, P < 0.01)$ .

(a). Explain why the Random and Independent conditions are met in this case.

(b) The distribution of individual responses is not Normal, because there is only a seven-point scale. Why is it still proper to use a two-sample t-test?

(c) Interpret the P-value in context.

Your teacher brings two bags of colored goldfish crackers to class. She tells you that Bag $1$ has $25 %$ red crackers and Bag $2$ has $35 %$ red crackers. Each bag contains more than $500$ crackers. Using a paper cup, your teacher takes an SRS of $50$ crackers from Bag 1 and a separate SRS of $40$ crackers from Bag $2$ . Let $\hat{p_{1}} - \hat{p_{2}}$ be the difference in the sample proportions of red crackers.

What is the shape of the sampling distribution of $\hat{p_{1}} - \hat{p_{2}}$ ? Why?

A random sample of 100 of a certain popular car model last year found that 20 had a certain minor defect in the brakes. The car company made an adjustment in the production process to try to reduce the proportion of cars with the brake problem. A random sample of 350 of this year's model found that 50 had the minor brake defect.

(a) Was the company's adjustment successful? Carry out an appropriate test to support your answer.

(b) Describe a Type I error and a Type Il error in this setting, and give al possible consequence of each.

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