The Practice of Statistics for AP

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

Chapter 8

Estimating with Confidence - all with Video Answers

Educators

JK

Section 1

Confidence Intervals: The Basics

Problem 1

In Exercises 1 to 4, determine the point estimator you would use and calculate the value of the point estimate.
Got shoes? How many pairs of shoes, on average, do female teens have? To find out, an AP Statistics class conducted a survey. They selected an SRS of 20 female students from their school. Then they
recorded the number of pairs of shoes that each student reported having. Here are the data:
$\begin{array}{llllllllll}{50} & {26} & {26} & {31} & {57} & {19} & {24} & {22} & {23} & {38} \\ {13} & {50} & {13} & {34} & {23} & {30} & {49} & {13} & {15} & {51} \\ \hline\end{array}$

Bryan Meares

Bryan Meares

Numerade Educator

Problem 2

In Exercises 1 to 4, determine the point estimator you would use and calculate the value of the point estimate.

Got shoes? The class in Exercise 1 wants to estimate the variability in the number of pairs of shoes that
female students have by estimating the population variance $\sigma^{2}$

Bryan Meares

Numerade Educator

Problem 3

In Exercises 1 to 4, determine the point estimator you would use and calculate the value of the point estimate.

Going to the prom Tonya wants to estimate what proportion of the seniors in her school plan to attend
the prom. She interviews an SRS of 50 of the 750 seniors in her school and finds that 36 plan to go to
the prom.

Bryan Meares

Numerade Educator

Problem 4

In Exercises 1 to 4, determine the point estimator you would use and calculate the value of the point estimate.

Reporting cheating What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of 172 undergraduates at a large university: “You witness two
students cheating on a quiz. Do you go to the professor?” Only 19 answered “Yes.”3

Bryan Meares

Numerade Educator

Problem 5

NAEP scores Young people have a better chance of full-time employment and good wages if they are
good with numbers. How strong are the quantitative skills of young Americans of working age? One source of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0 to 500. For example, a person who scores 233 can add the amounts of two checks appearing on a bank deposit slip; someone scoring 325 can determine the price of a meal from a menu; a person scoring 375 can transform a price in cents per ounce into dollars per pound. $^{4}$ Suppose that you give the NAEP test to an SRS of
840 people from a large population in which the scores have mean 280 and standard deviation S 60. The mean $\overline{x}$ of the 840 scores will vary if you take repeated samples.
(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x} .$
(b) Sketch the sampling distribution of $\overline{x}$ . Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the $68-95-99.7$ rule, about 95$\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m ?$ Shade the region on the axis of your sketch that is within $m$ of the mean.
(d) Whenever $\overline{x}$ falls in the region you shaded, the population mean $\mu$ lies in the confidence interval $\overline{x} \pm m .$ For what percent of all possible samples does the interval capture $\mu$ ?

Bryan Meares

Numerade Educator

Problem 6

Auto emissions Oxides of nitrogen (called NOX for short) emitted by cars and trucks are important contributors to air pollution. The amount of NOX emitted by a particular model varies from vehicle to
vehicle. For one light-truck model, NOX emissions vary with mean $\mu$ that is unknown and standard deviation $\sigma=0.4$ gram per mile. You test an SRS of 50 of these trucks. The sample mean NOX level $\overline{x}$ estimates the unknown $\mu .$ You will get different values of $\overline{x}$ if you repeat your sampling.
(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x} .$
(b) Sketch the sampling distribution of $\overline{x}$ . Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the $68-95-99.7$ rule, about 95$\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m ?$ Shade the region on the axis of your sketch that is within $m$ of the mean.
(d) Whenever $\overline{x}$ falls in the region you shaded, the unknown population mean $\mu$ lies in the confidence interval $\overline{x} \pm m$ . For what percent of all possible samples does the interval capture $\mu ?$

Bryan Meares

Numerade Educator

Problem 7

NAEP scores Refer to Exercise 5. Below your sketch, choose one value of $\overline{x}$ inside the shaded region and draw its corresponding confidence interval. Do the same for one value of $\overline{x}$ outside the shaded region. What is the most important difference between these
intervals? (Use Figure $8.5,$ on page $474,$ as a model for your drawing.)

Bryan Meares

Numerade Educator

Problem 8

Auto emissions Refer to Exercise $6 .$ Below your sketch, choose one value of $\overline{x}$ inside the shaded region and draw its corresponding confidence interval. Do the same for one value of $\overline{x}$ outside the shaded region. What is the most important difference between these intervals? (Use Figure $8.5,$ on page $474,$ as a model for your drawing.)

Bryan Meares

Numerade Educator

Problem 9

How confident? The figure below shows the result of taking 25 SRSs from a Normal population and
constructing a confidence interval for each sample. Which confidence level—80%, 90%, 95%, or 99%—
do you think was used? Explain.

Bryan Meares

Numerade Educator

Problem 10

How confident? The figure at top right shows the result of taking 25 SRSs from a Normal population
and constructing a confidence interval for each sample. Which confidence level—80%, 90%, 95%, or
99%—do you think was used? Explain.

Bryan Meares

Numerade Educator

Problem 11

Prayer in school A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes.” The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with 1,664 adults around the United States, excluding Alaska and Hawaii. . . . The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a 95% confidence level as 3 percentage points.
(a) Explain what the margin of error means to someone who knows little statistics.
(b) State and interpret the 95% confidence interval.
(c) Interpret the confidence level.

Sheryl Ezze

Numerade Educator

Problem 12

Losing weight A Gallup Poll in November 2008 found that 59% of the people in its sample said “Yes”
when asked, “Would you like to lose weight?” Gallup announced: “For results based on the total sample of national adults, one can say with 95% confidence that the margin of (sampling) error is $\pm 3$ percentage points." the margin of (sampling) error is $\pm 3$ percentage points."
(a) Explain what the margin of error means in this setting.
(b) State and interpret the 95% confidence interval.
(c) Interpret the confidence level.

Bryan Meares

Numerade Educator

Problem 13

Prayer in school Refer to Exercise 11. The news article goes on to say: “The theoretical errors do not
take into account a margin of additional error resulting from the various practical difficulties in taking any survey of public opinion.” List some of the “practical difficulties” that may cause errors in addition to the $\pm 3$ percentage point margin of error. Pay particular attention to the news article's description of the sampling method.

Bryan Meares

Numerade Educator

Problem 14

Losing weight Refer to Exercise 12. As Gallup indicates, the 3 percentage point margin of error for
this poll includes only sampling variability (what they call “sampling error”). What other potential sources
of error (Gallup calls these “nonsampling errors”) could affect the accuracy of the 59% estimate?

Bryan Meares

Numerade Educator

Problem 15

Shoes The AP Statistics class in Exercise 1 also asked an SRS of 20 boys at their school how many shoes they have. A 95% confidence interval for the difference in the population means (girls – boys) is
10.9 to 26.5. Interpret the confidence interval and the confidence level.

Bryan Meares

Numerade Educator

Problem 16

Lying online Many teens have posted profiles on sites such as Facebook and MySpace. A sample
survey asked random samples of teens with online profiles if they included false information in their
profiles. Of 170 younger teens (ages 12 to 14) polled, 117 said “Yes.” Of 317 older teens (ages 15
to 17) polled, 152 said “Yes.”6 A 95% confidence interval for the difference in the population proportions (younger teens – older teens) is 0.120 to 0.297. Interpret the confidence interval and the confidence level.

Sheryl Ezze

Numerade Educator

Problem 17

Explaining confidence $A 95 \%$ confidence interval for the mean body mass index (BMI) of young
American women is $26.8 \pm 0.6 .$ Discuss whether each of the following explanations is correct.
(a) We are confident that 95% of all young women have BMI between 26.2 and 27.4.
(b) We are 95% confident that future samples of young women will have mean BMI between 26.2 and
27.4.
(c) Any value from 26.2 to 27.4 is believable as the true mean BMI of young American women.
(d) In 95% of all possible samples, the population mean BMI will be between 26.2 and 27.4.
(e) The mean BMI of young American women cannot be 28.

Bryan Meares

Numerade Educator

Problem 18

Explaining confidence The admissions director from Big City University found that $(107.8,116.2)$ is a
95$\%$ confidence interval for the mean 1$Q$ score of all freshmen. Comment on whether or not each of the following explanations is correct.
(a) There is a 95$\%$ probability that the interval from 107.8 to 116.2 contains $\mu .$
(b) There is a 95$\%$ chance that the interval $(107.8,$ 116.2 ) contains $\overline{x}$ .
(c) This interval was constructed using a method that produces intervals that capture the true mean in 95$\%$ of all possible samples.
(d) 9$\%$ of all possible samples will contain the interval $(107.8,116.2) .$
(e) The probability that the interval $(107.8,116.2)$ captures $\mu$ is either 0 or $1,$ but we don't know which.

JK

Joshua Kinney

Numerade Educator

Problem 19

Conditions Explain briefly why each of the three conditions—Random, Normal, and Independent—is
important when constructing a confidence interval.

Bryan Meares

Numerade Educator

Problem 20

Plagiarizing An online poll posed the following question:
It is now possible for school students to log on to Internet sites and download homework. Everything
from book reports to doctoral dissertations can be downloaded free or for a fee. Do you believe that
giving a student who is caught plagiarizing an F for their assignment is the right punishment? Of the 20,125 people who responded, 14,793 clicked “Yes.” That’s 73.5% of the sample. Based on this sample, a 95% confidence interval for the percent of the population who would say "Yes" is $73.5 \% \pm$ 0.61%. Which of the three inference conditions is violated? Why is this confidence interval worthless?

Bryan Meares

Numerade Educator

Problem 21

Multiple choice: Select the best answer for Exercises 21 to 24.
A researcher plans to use a random sample of n 500 families to estimate the mean monthly family
income for a large population. A 99% confidence interval based on the sample would be ______ than a 90% confidence interval.
(a) narrower and would involve a larger risk of being incorrect
(b) wider and would involve a smaller risk of being incorrect
(c) narrower and would involve a smaller risk of being incorrect
(d) wider and would involve a larger risk of being incorrect
(e) wider, but it cannot be determined whether the risk of being incorrect would be larger or smaller

Bryan Meares

Numerade Educator

Problem 22

Multiple choice: Select the best answer for Exercises 21 to 24.
In a poll,
I. Some people refused to answer questions.
II. People without telephones could not be in the sample.
III. Some people never answered the phone in several calls.
Which of these sources is included in the $\pm 2 \%$ margin of error announced for the poll?
(a) I only (c) III only (e) None of these
(b) II only (d) I, II, and III

Bryan Meares

Numerade Educator

Problem 23

Multiple choice: Select the best answer for Exercises 21 to 24.
You have measured the systolic blood pressure of an SRS of 25 company employees. A 95% confidence interval for the mean systolic blood pressure for the employees of this company is (122, 138). Which of the following statements gives a valid interpretation of this interval?
(a) 95% of the sample of employees have a systolic blood pressure between 122 and 138.
(b) 95% of the population of employees have a systolic blood pressure between 122 and 138.
(c) If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure.
(d) The probability that the population mean blood pressure is between 122 and 138 is 0.95.
(e) If the procedure were repeated many times, 95% of the sample means would be between 122 and 138.

Bryan Meares

Numerade Educator

Problem 24

Multiple choice: Select the best answer for Exercises 21 to 24.
A polling organization announces that the proportion of American voters who favor congressional term limits is 64%, with a 95% confidence margin of error of 3%. If the opinion poll had announced the margin of error for 80% confidence rather than 95% confidence, this margin of error would be
(a) 3%, because the same sample is used.
(b) less than 3%, because we require less confidence.
(c) less than 3%, because the sample size is smaller.
(d) greater than 3%, because we require less confidence.
(e) greater than 3%, because the sample size is smaller.

Bryan Meares

Numerade Educator

Problem 25

Power lines and cancer (4.2, 4.3) Does living near power lines cause leukemia in children? The National Cancer Institute spent 5 years and $5 million gathering data on this question. The researchers compared 638 children who had leukemia with 620 who did not. They went into the homes and actually measured the magnetic fields in children’s bedrooms, in other rooms, and at the front door. They recorded
facts about power lines near the family home and also near the mother’s residence when she was pregnant. Result: no connection between leukemia and exposure to magnetic fields of the kind produced by power lines was found. $^{7}$
(a) Was this an observational study or an experiment? Justify your answer.
(b) Does this study show that living near power lines doesn’t cause cancer? Explain.

Bryan Meares

Numerade Educator

Problem 26

Sisters and brothers (3.1, 3.2) How strongly do physical characteristics of sisters and brothers
correlate? Here are data on the heights (in inches) of 11 adult pairs: $^{8}$
$$\begin{array}{llllllllll}\text{Brother:} \quad{71} & {68} & {66} & {67} & {70} & {71} & {70} & {73} & {72} & {65} & {66} \\ \text{Sister:}\quad\quad{69} & {64} & {65} & {63} & {65} & {62} & {65} & {64} & {66} & {59} & {62} \\ \hline\end{array}$$
(a) Construct a scatterplot using brother’s height as the explanatory variable. Describe what you see.
(b) Use your calculator to compute the least-squares regression line for predicting sister’s height from brother’s height. Interpret the slope in context.
(c) Damien is 70 inches tall. Predict the height of his sister Tonya.
(d) Do you expect your prediction in (c) to be very accurate? Give appropriate evidence to support your answer.

Sheryl Ezze

Numerade Educator