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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. AP3.3 (page 701)

Sports Illustrated planned to ask a random sample of Division I college athletes, “Do you believe performance-enhancing drugs are a problem in college sports?” Which of the following is the smallest number of athletes that must be interviewed to estimate the true proportion who believe performance-enhancing drugs are a problem within $\pm 2 %$ with $90 %$ confidence?
$a . 17 b . 21 c . 1680 d . 1702 e . 2401$

Short Answer

Expert verified

The correct optionis

a. $n = \frac{{[z_{α / 2}]}^{2} 0.25}{M E^{2}} = \frac{1 . 65^{2} \times 0.25}{0 . 02^{2}} \approx 1702$

Step by step solution

01

Given information

We have to find the smallest number of athletes that must be interviewed to estimate the true proportion.

02

Explanation

Formula sample size:

$\hat{p} known: n = \frac{{[z_{α / 2}]}^{2} \hat{p} \hat{q}}{M E^{2}} = \frac{{[z_{α / 2}]}^{2} \hat{p} (1 - \hat{p})}{M E^{2}}$

$\hat{p} unknown: n = \frac{{[z_{α / 2}]}^{2} 0.25}{M E^{2}}$

the sample size is:

$n = \frac{{[z_{α / 2}]}^{2} 0.25}{M E^{2}} = \frac{1 . 65^{2} \times 0.25}{0 . 02^{2}} \approx 1702$

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Most popular questions from this chapter

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 large toy company introduces many new toys to its product line each year. The

company wants to predict the demand as measured by y, first-year sales (in millions of dollars) using x, awareness of the product (as measured by the percent of customers who had heard of the product by the end of the second month after its introduction). A random sample of $65$ new products was taken, and a correlation of $0.96$ was computed. Which of the following is true?

a. The least-squares regression line accurately predicts first-year sales 96% of the time.

b. About $92 %$ of the time, the percent of people who have heard of the product by the end of the second month will correctly predict first-year sales.

c. About $92 %$ of first-year sales can be accounted for by the percent of people who have heard of the product by the end of the second month.

d. For each increase of 1% in awareness of the new product, the predicted sales will go up by $0.96$ million dollars.

e. About $92 %$ of the variation in first-year sales can be accounted for by the leastsquares regression line with the percent of people who have heard of the product by the end of the second month as the explanatory variable.

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.

Shortly before the $2012$ presidential election, a survey was taken by the school newspaper at a very large state university. Randomly selected students were asked, “Whom do you plan to vote for in the upcoming presidential election?” Here is a two-way table of the responses by political persuasion for $1850$ students:

Candidate of choice	Political persuasion
		Democrat	Republican	Independent	Total
	Obama	$925$	$78$	$26$	$1029$
	Romney	$78$	$598$	$19$	$695$
	Other	$2$	$8$	$11$	$21$
	Undecided	$32$	$28$	$45$	$105$
	Total	$1037$	$712$	$101$	$1850$

Which of the following statements about these data is true?

a. The percent of Republicans among the respondents is $41 % .$

b. The marginal relative frequencies for the variable choice of candidate are given by

Obama: $55.6 %$ ; Romney: $37.6 %$ ; Other: $1.1 %$ ; Undecided: $5.7 %$ .

c. About $11.2 %$ of Democrats reported that they planned to vote for Romney.

d. About $44.6 %$ of those who are undecided are Independents.

e. The distribution of political persuasion among those for whom Romney is the

candidate of choice is Democrat: $7.5 %$ ; Republican: $84.0 %$ ; Independent: $18.8 %$ .

Response bias Does the appearance of the interviewer influence how people respond to a survey question? Ken (white, with blond hair) and Hassan (darker, with Middle Eastern features) conducted an experiment to address this question. They took turns (in a random order) walking up to people on the main street of a small town, identifying themselves as students from a local high school, and asking them, “Do you support President Obama’s decision to launch airstrikes in Iraq?” Of the $50$ people Hassan spoke to, $11$ said “Yes,” while $21$ of the $44$ people Ken spoke to said “Yes.” Construct and interpret a $90 %$ confidence interval for the difference in the proportion of people like these who would say they support President Obama’s decision when asked by Hassan versus when asked by Ken.

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