Introduction to Probability and Statistics

William Mendenhall, III, Robert J. Beaver, Barbara M. Beaver

Chapter 7

Sampling Distributions - all with Video Answers

Educators

Section 1

Sampling Plans and Experimental Designs

Problem 1

Select $n=20$ experimental units from a population of size $N=500$. (HINT: Since you need to use threedigit numbers, you can assign 2 three-digit numbers to each of the experimental units. For example, unit 1 would correspond to random numbers 001 and 501 , unit 2 would correspond to 002 and $502,$ and so on.)

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 2

Select $n=20$ people from a population of size $N=2000$ for a political opinion poll.

Joshua Argo

Numerade Educator

Problem 3

Select $n=15$ voters from a population of 50,000 voters.

Raymond Matshanda

Raymond Matshanda

Numerade Educator

Problem 4

Select $n=10$ drivers from a DMV database containing 20,000 drivers.

Jon Southam

Numerade Educator

Problem 5

Ten homes are randomly selected and all adult occupants are surveyed.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 6

An auditor selects every 100 th entry in a ledger for amount verification.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 7

A store manager randomly selects 10 sales receipts from each of the store's six departments.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 8

A doctor chooses six of his most "at risk" patients to participate in a clinical trial.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 9

A student waits until Sunday night to complete a survey for his sociology class. He asks 10 of his fraternity brothers to participate.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 10

A population consists of $40 \%$ women and $60 \%$ men. The researcher decides to choose a sample consisting of 20 women and 30 men.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 11

A professor chooses the 20 students in his class whom he thinks will most likely return his questionnaire.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 12

A random sample of $n=50$ city blocks is selected, and a census is done for each single-family dwelling on each block.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 13

The highway patrol stops every 10th vehicle on a given city street between 9: 00 a.m. and 3: 00 p.m. to perform a routine traffic safety check

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 14

One hundred households in each of four city wards are surveyed concerning a pending city tax relief referendum.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 15

Every 10th tree in a managed slash pine plantation is checked for pine needle borer infestation.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 16

A random sample of $n=1000$ taxpayers from the city of San Bernardino is selected by the Internal Revenue Service and their tax returns are audited.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 17

Only $30 \%$ of a mailed survey are returned. However, over $90 \%$ of the surveys returned agree with a proposed zoning change.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 18

A question on a voter survey asked, "Do you agree that the current administration is 'soft on crime'?"

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 19

You post a short survey on Facebook and publish the results based on the reported responses.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 20

A random sample of public opinion in a small town was obtained by selecting every 10 th person who passed by the busiest corner in the downtown area. Will this sample have the characteristics of a random sample selected from the town's citizens? Explain.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 21

A questionnaire was mailed to 1000 registered municipal voters selected at random. Only 500 questionnaires were returned, and of the 500 returned, 360 respondents were strongly opposed to a surcharge proposed to support the city Parks and Recreation Department. Are you willing to accept the $72 \%$ figure as a valid estimate of the percentage in the city who are opposed to the surcharge? Why or why not?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 22

In many states, lists of possible jurors are assembled from voter registration lists and Department of Motor Vehicles records of licensed drivers and car owners. In what ways might this list not cover certain sectors of the population adequately?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 23

One question on a survey questionnaire is phrased as follows: "Don't you agree that there is too much sex and violence during prime TV viewing hours?" Comment on possible problems with the responses to this question. Suggest a better way to pose the question.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 24

New research shows that omega-3 fats may not help reduce second heart attacks in heart attack survivors. The study included 4837 men and women being treated for heart disease. The experimental group received an additional $400 \mathrm{mg}$ of the fats daily. ' Suppose that this experiment was repeated with 50 individuals in the control group and 50 individuals in the experimental group. Provide a randomization scheme to assign the 100 individuals to the two groups.

Erin Moser

Numerade Educator

Problem 25

Does the race of an interviewer matter? This question was investigated and reported in an issue of Chance magazine. $^{2}$ The interviewer asked, "Do you feel that affirmative action should be used as an occupation selection criteria?" with possible answers of
yes or no.
a. What problems might you expect with responses to this question when asked by interviewers of different ethnic origins?
b. When people were interviewed by an African American, the response was about $70 \%$ in favor of affirmative action, approximately $35 \%$ when interviewed by an Asian, and approximately $25 \%$ when interviewed by a Caucasian. Do these results support your answer in part a?

Victor Salazar

Numerade Educator

Problem 26

In a Fox News Poll $^{3}$ conducted by Pulse Opinion Research in the state of Delaware, 1000 likely voters were asked to answer the following questions.
- All in all, would you rather have bigger government that provides more services or smaller government that provides fewer services?
- Do you agree or disagree with the following statement: The federal government has gotten totally out of control and threatens our basic liberties
unless we clear house and commit to drastic
change.
- Thinking about the health care law that was passed earlier this year, would you favor repealing the new law to keep it from going into effect, or would you oppose repealing the new law?
Comment on the effect of wording bias on the responses gathered using this survey.

Emily Himsel

Numerade Educator

Problem 27

A small new study shows that tai chi, an ancient Chinese practice of exercise and meditation, may relieve symptoms of chronic painful fibromyalgia. The study assigned 66 fibromyalgia patients to take either a 12 -week tai chi class, or attend a wellness education class.
a. Provide a randomization scheme to assign 66 subjects to the two groups.
b. Will your randomization scheme result in equal sized groups? Explain.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 28

Two different Gallup Polls were conducted for CNN/USA Today, both of which involved people's feelings about the U.S. space program. ${ }^{5}$ Here is a question from each poll, along with the responses of the sampled Americans:
a. Read the two poll questions. Which of the two wordings is more unbiased? Explain.
b. Look at the responses for the two different polls. How would you explain the large differences in the percentages either favoring or opposing the new program?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 29

The battle for consumer preference continues between Pepsi and Coke. How can you weigh in? There is a website where you can vote for one of these colas if you click on the link that says PAY CASH for your opinion. Explain why the respondents do not represent a random sample of the opinions of purchasers or drinkers of these drinks. Explain the types of distortions that could creep into an Internet opinion poll.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 30

A research psychologist is planning an experiment to determine whether the use of imagery - picturing a word in your mind - affects people's ability to memorize. He wants to use two groups of subjects: a group that memorizes a set of 20 words using the imagery technique, and a control group that does not use imagery.
a. Use a randomization technique to divide a group of 20 subjects into two groups of equal size.
b. How can the researcher randomly select the group of 20 subjects?
c. Suppose the researcher offers to pay subjects $\$ 50$ each to participate in the experiment and uses the first 20 students who apply. Would this group behave as if it were a simple random sample of size $n=20 ?$

Kimberly Waterbury

Kimberly Waterbury

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