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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. 31 (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� The $95 %$ confidence interval for the difference $p_{M} - p_{F}$ in the proportions of college men and women who worked last summer is about
(a) $0.06 卤 0.00095$
(b) $0.06 卤 0.043 .$
(c) $0.06 卤 0.036$
(d) $- 0.06 卤 0.043$
(e) $- 0.06 卤 0.036$

Short Answer

Expert verified

The difference in the proportions of college men and women who worked last summer is option (b) $0.06 卤 0.043 .$

Step by step solution

01

Given information

The number of women who worked last summer $= 410$

The number of men who worked last summer $= 484$

Find the difference in the proportions of college men and women who worked last summer.

02

Explanation

The given data is written as

$x_{1} = 484$

$n_{1} = 550$

$x_{2} = 410$

$n_{2} = 500$

By dividing the number of successes by the sample size, the sample proportion is calculated:

localid="1650360665085" ${\hat{p}}_{1} = \frac{x_{1}}{n_{1}} = \frac{484}{550} = 0.88$

localid="1650360679467" ${\hat{p}}_{2} = \frac{x_{2}}{n_{2}} = \frac{410}{500} = 0.82$

For confidence level $1 - 伪 = 0.95$ ,

find $z_{伪 / 2} = z_{0.025}$ using table II

$z_{伪 / 2} = 1.96$

The difference between the endpoints of the confidence $p_{1} - p_{2}$ are:

localid="1650360715016" $= ({\hat{p}}_{1} - {\hat{p}}_{2}) 卤 z_{伪 / 2} 路 \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}} = (0.88 - 0.82) 卤 1.96 \sqrt{\frac{0.88 (1 - 0.88)}{550} + \frac{0.82 (1 - 0.82)}{500}}$

$鈮� 0.06 卤 0.043 .$

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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 $x_{1} - {\overset{炉}{x}}_{2}$ be the difference in the sample mean amount of liquid under the two settings (large 鈥� medium). Find the probability that ${\overset{炉}{x}}_{1} - {\overset{炉}{x}}_{2}$ is more than $12$ ounces. Show your work.

Find the probability that $\hat{p_{1}} - \hat{p_{2}}$ is less than or equal to $0.02$ . Show your work.

鈥淲ould you marry a person from a lower social class than your own?鈥� Researchers asked this question of a random sample of $385$ black, never married students at two historically black colleges in the South. Of the $149$ men in the sample, $91$ said 鈥淵es.鈥� Among the $236$ women, $117$ said 鈥淵es.鈥� $14$ Is there reason to think that different proportions of men and women in this student population would be willing to marry beneath their class?

Holly carried out the significance test shown below to answer this question. Unfortunately, she made some mistakes along the way. Identify as many mistakes as you can, and tell how to correct each one.

State: I want to perform a test of

$H_{0} : p_{1} = p_{2}$

$H_{a} : p_{1} 鈮� p_{2}$

at the $95 %$ confidence level.

Plan: If conditions are met, I鈥檒l do a one-sample $z$ test for comparing two proportions.

Random The data came from a random sample of 385 black, never-married students.
Normal One student鈥檚 answer to the question should have no relationship to another student鈥檚 answer.
Independent The counts of successes and failures in the two groups $鈥� 91, 58, 117$ , and $119 鈥�$ are all at least $10$

Do: From the data, ${\hat{p}}_{1} = \frac{91}{149} = 0.61$ and ${\hat{p}}_{2} = \frac{117}{236} = 0.46$ .

Test statistic

$z = \frac{(0.61 - 0.46) - 0}{\sqrt{\frac{0.61 (0.39)}{149} + \frac{0.46 (0.54)}{236}}} = 2.91$

$p$ =value From Table A, role="math" localid="1650292307192" $P (z 鈮� 2.91) 鈮� 1 - 0.3982 鈮� 0.0018$ .

Conclude: The $p$ -value, $0.0018$ , is less than $0.05$ , so I鈥檒l reject the null hypothesis. This proves that a higher proportion of men than women are willing to marry someone from a social class lower than their own.

How quickly do synthetic fabrics such as polyester decay in landfills? A researcher buried polyester strips in the soil for different lengths of time, then dug up the strips and measured the force required to break them. Breaking strength is easy to measure and is a good indicator of decay. Lower strength means the fabric has decayed. For one part of the study, the researcher buried $10$ strips of polyester fabric in well-drained soil in the summer. The strips were randomly assigned to two groups: $5$ of them were buried for $2$ weeks and the other $5$ were buried for $16$ weeks. Here are the breaking strengths in pounds

Do the data give good evidence that polyester decays more in $16$ weeks than in $2$ weeks? Carry out an appropriate test to help answer this question .

Refer to Exercise $16$ .

(a) Carry out a significance test at the $伪 = 0.05$ level.

(b) Construct and interpret a $95 %$ confidence interval for the difference between the population proportions. Explain how the confidence interval is consistent with the results of the test in part (a).

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