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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 10: Q 8. (page 640)

Digital video disks A company that records and sells rewritable DVDs wants to compare the reliability of DVD fabricating machines produced by two different manufacturers. They randomly select $500$ DVDs produced by each fabricator and find that $484$ of the disks produced by the first machine are acceptable and $480$ of the disks produced by the second machine are acceptable. If $p_{1}, p_{2}$ are the proportions of acceptable DVDs produced by the first and second machines, respectively, check if the conditions for calculating a confidence interval for $p_{1} - p_{2}$ are met.

Short Answer

Expert verified

All conditions are satisfied, we can find confidence interval.

Step by step solution

01

Given Information

It is given that $n_{1} = 500$

$x_{1} = 484$

$n_{2} = 500$

$x_{2} = 480$

02

Explanation

Three conditions for hypothesis test are:

Random: As assigned randomly, it is satisfied.

Independent: As all data values in population and used sample, it is satisfied.

Normal: Success in first sample is $484$ , and failures are $500 - 484 = 16$ , both are greater than ten , it is also satisfied.

All conditions are met, we can find confidence interval for $p_{1} - p_{2}$

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Most popular questions from this chapter

Children make choices Refer to Exercise 15.

a. Explain why the sample results give some evidence for the alternative hypothesis.

b. Calculate the standardized test statistic and $P$ -value.

c. What conclusion would you make?

Literacy A researcher reports that $80 %$ of high school graduates, but only $40 %$ of high school dropouts, would pass a basic literacy test. Assume that the researcher’s claim is true. Suppose we give a basic literacy test to a random sample of $60$ high school graduates and a separate random sample of $75$ high school dropouts. ${\hat{p}}_{G}, {\hat{p}}_{D}$ be the sample proportions of graduates and dropouts, respectively, who pass the test.

a. What is the shape of the sampling distribution of ${\hat{p}}_{G} - {\hat{p}}_{D}$ . Why?

b. Find the mean of the sampling distribution.

c. Calculate and interpret the standard deviation of the sampling distribution.

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of $60$ high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be $6$ hours with a standard deviation of $3$ hours. The researcher also obtained an independent SRS of $40$ high school students in a large city school district and found the mean time spent in extracurricular activities per week to be $5$ hours with a standard deviation of $2$ hours. Suppose that the researcher decides to carry out a significance test of $H_{0}$ : μsuburban=μcity versus a two-sided alternativ

The P-value for the test is $0.048$ . A correct conclusion is to

a. fail to reject $H_{0}$ because $0.048 < α = 0.05$ . There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

b. fail to reject $H_{0}$ because $0.048 < α = 0.05$ . There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

c. fail to reject $H_{0}$ because $0.048 < α = 0.05$ . There is convincing evidence that the average time spent on extracurricular activities by students in the suburban and city school districts is the same.

d. reject $H_{0}$ because $0.048 < α = 0.05$ . There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

e. reject $H_{0}$ because $0.048 < α = 0.05$ . There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

Bag lunch? Phoebe has a hunch that older students at her very large high

school are more likely to bring a bag lunch than younger students because they have grown tired of cafeteria food. She takes a simple random sample of 80 sophomores and finds that $52$ of them bring a bag lunch. A simple random sample of $104$ seniors reveals that $78$ of them bring a bag lunch.

a. Do these data give convincing evidence to support Phoebe’s hunch at the $α = 0.05$ significance level?

b. Interpret the $P$ -value from part (a) in the context of this study.

Don’t drink the water!The movie A Civil Action ( $1998$ ) tells the story of a

major legal battle that took place in the small town of Woburn, Massachusetts. A town well that supplied water to east Woburn residents was contaminated by industrial chemicals. During the period that residents drank water from this well, $16$ of $414$ babies born had birth defects. On the west side of Woburn, $3$ of $228$ babies born during the same time period had birth defects. Let $p_{1}$ be

the true proportion of all babies born with birth defects in west Woburn and $p_{2}$ be the true proportion of all babies born with birth defects in east Woburn. Check if the conditions for calculating a confidence interval for $p_{1} - p_{2}$ are met.

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