The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore

Chapter 7

Sampling Distributions - all with Video Answers

Educators

Section 1

What Is a Sampling Distribution?

Problem 1

Healthy living
(a) A random sample of 1000 people who signed a card saying they intended to quit smoking were contacted 9 months later. It turned out that $210(21 \%)$ of the sampled individuals had not smoked over the past 6 months.
(b) Tom is cooking a large turkey breast for a holiday meal. He wants to be sure that the turkey is safe to eat, which requires a minimum internal temperature of $165^{\circ} \mathrm{F}$. Tom uses a thermometer to measure the temperature of the turkey meat at four randomly chosen points. The minimum reading in the sample is $170^{\circ} \mathrm{F}$.

James Kiss

James Kiss

Numerade Educator

Problem 2

The economy
(a) Each month, the Current Population Survey interviews a random sample of individuals in about 60,000
U.S. households. One of their goals is to estimate the national unemployment rate. In October $2012,7.9 \%$ of those interviewed were unemployed.
(b) How much do gasoline prices vary in a large city? To find out, a reporter records the price per gallon of regular unleaded gasoline at a random sample of 10 gas stations in the city on the same day. The range (maximum minimum) of the prices in the sample is 25 cents.

James Kiss

Numerade Educator

Problem 3

Get your bearings A large container is full of ball bearings with mean diameter 2.5003 centimeters $(\mathrm{cm})$. This is within the specifications for acceptance of the container by the purchaser. By chance, an inspector chooses 100 bearings from the container that have mean diameter $2.5009 \mathrm{~cm}$. Because this is outside the specified limits, the container is mistakenly rejected.

James Kiss

Numerade Educator

Problem 4

Voters Voter registration records show that $41 \%$ of voters in a state are registered as Democrats. To test a random digit dialing device, you use it to call 250 randomly chosen residential telephones in the state. Of the registered voters contacted, $33 \%$ are registered Democrats.

James Kiss

Numerade Educator

Problem 5

Unlisted numbers A telemarketing firm in a large city uses a device that dials residential telephone numbers in that city at random. Of the first 100 numbers dialed, $48 \%$ are unlisted. This is not surprising because $52 \%$ of all residential phones in the city are unlisted.

James Kiss

Numerade Educator

Problem 6

How tall? A random sample of female college students has a mean height of 64.5 inches, which is greater than the 63 -inch mean height of all adult American women.

James Kiss

Numerade Educator

Problem 7

Sampling distribution
(a) List all 15 possible SRSs of size $n=2$ from the population. Find the value of $\bar{x}$ for each sample.
(b) Make a graph of the sampling distribution of $\bar{x}$. Describe what you see.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 8

Sampling distribution
(a) List all 15 possible SRSs of size $n=2$ from the population. Find the value of the range for each sample.
(b) Make a graph of the sampling distribution of the sample range. Describe what you see.

Jon Southam

Numerade Educator

Problem 9

Doing homework A school newspaper article claims that $60 \%$ of the students at a large high school did all their assigned homework last week. Some skeptical $\mathrm{AP}^{\otimes}$ Statistics students want to investigate whether this claim is true, so they choose an SRS of 100 students from the school to interview. What values of the sample proportion $\hat{p}$ would be consistent with the claim that the population proportion of students who completed all their homework is $p=0.60 ?$ To find out, we used Fathom software to simulate choosing 250 SRSs of size $n=100$ students from a population in which $p=0.60$. The figure below is a dotplot of the sample proportion $\hat{p}$ of students who did all their homework.
(a) There is one dot on the graph at $0.73 .$ Explain what this value represents.
(b) Describe the distribution. Are there any obvious outliers?
(c) Would it be surprising to get a sample proportion of 0.45 or lower in an $\mathrm{SRS}$ of size 100 when $p=0.6 ?$ Justify your answer.
(d) Suppose that 45 of the 100 students in the actual sample say that they did all their homework last week. What would you conclude about the newspaper article's claim? Explain.

Teresa Wray

Numerade Educator

Problem 10

Tall girls According to the National Center for Health Statistics, the distribution of heights for 16-year-old females is modeled well by a Normal density curve with mean $\mu=64$ inches and standard deviation $\sigma=2.5$ inches. To see if this distribution applies at their high school, an $\mathrm{AP}^{\otimes}$ Statistics class takes an SRS of 20 of the 300 16-year-old females at the school and measures their heights. What values of the sample mean $\bar{x}$ would be consistent with the population distribution being $N(64,2.5) ?$ To find out, we used Fathom software to simulate choosing 250 SRSs of size $n=20$ students from a population that is $N(64,$ 2.5). The figure below is a dotplot of the sample mean height $\bar{x}$ of the students in each sample.
(a) There is one dot on the graph at $62.4 .$ Explain what this value represents.
(b) Describe the distribution. Are there any obvious outliers?
(c) Would it be surprising to get a sample mean of 64.7 or more in an SRS of size 20 when $\mu=64 ?$ Justify
your answer.
(d) Suppose that the average height of the 20 girls in the class's actual sample is $\bar{x}=64.7$. What would you conclude about the population mean height $\mu$ for the 16 -year-old females at the school? Explain.

Bryan Meares

Numerade Educator

Problem 11

Doing homework Refer to Exercise 9 .
(a) Make a bar graph of the population distribution given that the newspaper's claim is correct.
(b) Sketch a possible graph of the distribution of sample data for the SRS of size 100 taken by the $\mathrm{AP}^{\otimes}$ Statistics students.

Sheryl Ezze

Numerade Educator

Problem 12

Tall girls Refer to Exercise l0.
(a) Make a graph of the population distribution.
(b) Sketch a possible dotplot of the distribution of sample data for the SRS of size 20 taken by the $\mathrm{AP}^{\otimes}$ Statistics class.

Shu Naito

Numerade Educator

Problem 13

Cold cabin? The Fathom screen shot below shows the results of taking 500 SRSs of 10 temperature readings from a population distribution that is $N(50,3)$ and recording the sample variance $s_{x}^{2}$ each time.
(a) Describe the approximate sampling distribution.
(b) Suppose that the variance from an actual sample is $s_{x}^{2}=25 .$ What would you conclude about the thermostat manufacturer's claim? Explain.

Teresa Wray

Numerade Educator

Problem 14

Cold cabin? The Fathom screen shot below shows the results of taking 500 SRSs of 10 temperature readings from a population distribution that is $N(50,3)$ and recording the sample minimum each time.
(a) Describe the approximate sampling distribution.
(b) Suppose that the minimum of an actual sample is $40^{\circ} \mathrm{F}$. What would you conclude about the thermostat manufacturer's claim? Explain.

Teresa Wray

Numerade Educator

Problem 15

A sample of teens A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of 13 - to 16 -year-olds. The researchers will report the mean $\bar{x}$ from their sample as an estimate of the mean cholesterol level $\mu$ in this population. Explain to someone who knows little about statistics what it means to say that $\bar{x}$ is an unbiased estimator of $\mu$.

James Kiss

Numerade Educator

Problem 16

Predict the election A polling organization plans to ask a random sample of likely voters who they plan to vote for in an upcoming election. The researchers will report the sample proportion $\hat{p}$ that favors the incumbent as an estimate of the population proportion $p$ that favors the incumbent. Explain to someone who knows little about statistics what it means to say that $\hat{p}$ is an unbiased estimator of $p$.

James Kiss

Numerade Educator

Problem 17

A sample of teens Refer to Exercise $15 .$ The sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$ no matter what size SRS the study chooses. Explain to someone who knows nothing about statistics why a large random sample will give more trustworthy results than a small random sample.

James Kiss

Numerade Educator

Problem 18

Predict the election Refer to Exercise l6. The sample proportion $\hat{p}$ is an unbiased estimator of the population proportion $p$ no matter what size random sample the polling organization chooses. Explain to someone who knows nothing about statistics why a large random sample will give more trustworthy results than a small random sample.

James Kiss

Numerade Educator

Problem 19

Bias and variability The figure below shows histograms of four sampling distributions of different statistics intended to estimate the same parameter.
(a) Which statistics are unbiased estimators? Justify your answer.
(b) Which statistic does the best job of estimating the parameter? Explain.

Jacquelinne S. Mejia Sandoval

Jacquelinne S. Mejia Sandoval

Numerade Educator

Problem 20

IRS audits The Internal Revenue Service plans to examine an SRS of individual federal income tax returns. The parameter of interest is the proportion of all returns claiming itemized deductions. Which would be better for estimating this parameter: an SRS of 20,000 returns or an SRS of 2000 returns? Justify your answer.

James Kiss

Numerade Educator

Problem 21

At a particular college, $78 \%$ of all students are receiving some kind of financial aid. The school newspaper selects a random sample of 100 students and $72 \%$ of the respondents say they are receiving some sort of financial aid. Which of the following is true?
(a) $78 \%$ is a population and $72 \%$ is a sample.
(b) $72 \%$ is a population and $78 \%$ is a sample.
(c) $78 \%$ is a parameter and $72 \%$ is a statistic.
(d) $72 \%$ is a parameter and $78 \%$ is a statistic.
(e) $78 \%$ is a parameter and 100 is a statistic.

James Kiss

Numerade Educator

Problem 22

A statistic is an unbiased estimator of a parameter when
(a) the statistic is calculated from a random sample.
(b) in a single sample, the value of the statistic is equal to the value of the parameter.
(c) in many samples, the values of the statistic are very close to the value of the parameter.
(d) in many samples, the values of the statistic are centered at the value of the parameter.
(e) in many samples, the distribution of the statistic has a shape that is approximately Normal.

James Kiss

Numerade Educator

Problem 23

In a residential neighborhood, the median value of a house is 200,000 . For which of the following sample sizes is the sample median most likely to be above 250,000 ?
$\begin{array}{ll}\text { (a) } & n=10\end{array}$
(b) $n=50$
$\begin{array}{ll}\text { (c) } & n=100\end{array}$
(d) $n=1000$
(e) Impossible to determine without more information.

James Kiss

Numerade Educator

Problem 24

Increasing the sample size of an opinion poll will reduce the
(a) bias of the estimates made from the data collected in the poll.
(b) variability of the estimates made from the data collected in the poll.
(c) effect of nonresponse on the poll.
(d) variability of opinions in the sample.
(e) variability of opinions in the population.

James Kiss

Numerade Educator

Problem 25

Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose osteoporosis, an elaborate apparatus measures bone mineral density (BMD). BMD is usually reported in standardized form. The standardization is based on a population of healthy young adults. The World Health Organization (WHO) criterion for osteoporosis is a BMD score that is 2.5 standard deviations below the mean for young adults. BMD measurements in a population of people similar in age and gender roughly follow a Normal distribution.
(a) What percent of healthy young adults have osteoporosis by the $\mathrm{WHO}$ criterion?
(b) Women aged 70 to 79 are, of course, not young adults. The mean BMD in this age group is about
-2 on the standard scale for young adults. Suppose that the standard deviation is the same as for young adults. What percent of this older population has osteoporosis?

James Kiss

Numerade Educator

Problem 26

Squirrels and their food supply (3.2) Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over 16 years. ${ }^{3}$ Computer output from a leastsquares regression on these data and a residual plot are shown below.
(a) Give the equation for the least-squares regression line. Define any variables you use.
(b) Is a linear model appropriate for these data? Explain.
(c) Interpret the values of $r^{2}$ and $s$ in context.

James Kiss

Numerade Educator