AP Statistics Chapter 4 Homework Packet Introduction to ... working on the project ... Instant...

AP Statistics Chapter 4 Homework Packet

Introduction to Sampling (4.1A) Name_____________________ Hr____

1. Students as customers The Mentor plans to survey local businesses about the

importance of students as customers. From telephone book listings, the Mentor

staff chooses 150 businesses at random. Of these, 73 return the questionnaire

mailed by the staff. Identify the population and the sample.

Population: Sample:

2. Student archaeologists An archaeological dig turns up large numbers of pottery

shards, broken stone tools, and other artifacts. Students working on the project

classify each artifact and assign it a number. The counts in different categories are

important for understanding the site, so the project director chooses 2% of the

artifacts at random and checks the students’ work. Identify the population and the

sample.

Population: Sample:

3. Call the shots A newspaper advertisement for an upcoming TV show said:

“Should handgun control be tougher? You call the shots in a special poll tonight.

Text YES or NO to 1-900-720-6181. Texting charges may apply.” Explain why

this opinion poll is almost certainly biased.

4. Explain it to the congresswoman You are on the staff of a member of Congress

who is considering a bill that would provide government-sponsored insurance for

nursing-home care. You report that 1128 letters have been received on the issue,

of which 871 oppose the legislation. “I’m surprised that most of my constituents

oppose the bill. I thought it would be quite popular,” says the congresswoman.

Are you convinced that a majority of the voters oppose the bill? How would you

explain the statistical issue to the congresswoman?

5. Instant opinion A recent online poll posed the question “should female athletes

be paid the same as men for the work they do?” In all, 13,147 (44%) said “Yes,”

15,182 (50%) said “No,” and the remaining 1448 said “Don’t know.” In spite of

the large sample size for this survey, we can’t trust the result. Why not?


6. Online polls In June 2008, Parade magazine posed the following question:

“Should drivers be banned from using all cell phones?” Readers were encouraged

to vote online at parade.com. The July 13, 2008, issue of Parade reported the

results: 2407 (85%) said “Yes” and 410 (15%) said “No.”

(a) What type of sample did the Parade survey obtain?

(b) Explain why this sampling method is biased. Is 85% probably higher or

lower than the true percent of all adults who believe that cell phone use

while driving should be banned? Why?

7. Do you trust the internet? You want to ask a sample of MHS students the

question “How much do you trust information about health that you find on the

Internet – a great deal, somewhat, not much, or not at all?” You try out this and

other questions on a pilot group of 5 students chosen from your class. The class

members are listed below.

(a) Explain how you would use a line of Table D to choose an SRS of 5

students from the following list. Explain your method clearly enough for a

classmate to obtain your results.

(b) Use line 107 to select the sample. Show how you use each of the digits.

Anderson Deng Glaus Nguyen Samuels

Arroyo De Ramos Helling Palmiero Shen

Batista Drasin Husain Percival Suh

Bell Fox Johnson Richards Washburn

Burke Gandhi Martin Rider Zabidi

Cabrera Garcia Murphy Rodriguez Zhang


8. Apartment Living You are planning a report on apartment living in a

Manhattan, KS. You decide to select three apartment complexes at random for

in-depth interviews with residents.

a) Explain how you would use a line of Table D to choose an SRS of 3

complexes from the list below. Explain your method clearly.

b) Use line 117 to select the sample.

Aggie Village Gaslight Village Pheasant Hill

Campus East Georgetown Park Place

Chase Heritage Ridge Plaza West

Crestwood Lamplighter University Gardens

Evergreen Manhattan Ave. Westchester

Founder’s Hill Pebblebrook Winston Place

Other Sampling Methods (4.1B)

1. Suppose 1000 iPhones are produced at a factory today. Management would like

to ensure that the phones’ display screens meet their quality standards before

shipping them to retail stores. Since it takes about 10 minutes to inspect each

individual screen, managers decide to inspect a sample of 20 phones from the

day’s production.

(a) Explain why it would be difficult for managers to inspect an SRS of 20

phones from the day’s production.

(b) An eager employee suggests that it would be easy to inspect the last 20

iPhones that were produced today. Why is this a bad idea?

(c) Another employee recommends inspecting every 50th

iPhone produced.

Explain why this is not an SRS.


2. On the west side of City Park, many mature pine trees are dying due to

infestation by Super Beetles. KSU scientists would like to use sampling to

estimate the proportion of all pine trees in the area that have been infected.

(a) Explain why an SRS is not practical in this setting.

(b) A possible alternative would be to use every pine tree along Poyntz Ave. as a

sample. Why is this sampling method biased?

(c) Suppose that a well-designed random sampling plan is carried out, and that

35% of the pine trees in City Park are infested by the Super Beetles. Can the

scientists conclude that 35% of all the pine trees in City Park are infested?

3. A club has 30 student members and 10 faculty members. The students are:

Barragar Glessner Kelly Liu Sobering

Booth Hart Klug Miller Wang

Borgarelli Hayes Knudsen Myers Weber

Burton Herrman Kohlmeier Ramaswamy Wefald

Calvert Hodges Larson Richt Weiss

Faucett Keck Lee Singh White

The faculty are:

Ackerman Dawdy Lamb Pape Schwandt

Braxmeyer Fink Mamolo Roberson Thies

Brooks Johnston Nelson Sapp Wichman

The club can send 5 students and 3 faculty members to a convention. It decides to

choose those who will go by random selection.

(a) How will you label the two strata? Use Table D, beginning at line 123 to

choose a stratified random sample of 5 students and 3 faculty members.

(b) What would be the problem with selecting a simple random sample to make

your selection?

http://www.bing.com/images/search?q=cartoon+bugs&view=detail&id=72DCD1CEA922889EA70BA9DD342985067D0B8EB8&first=91&qpvt=cartoon+bugs&FORM=IDFRIR


4. A hotel has 30 floors with 40 rooms per floor. The rooms on one side of the hotel

have a waterfront view, while rooms on the other side face a golf course. Of

course, there is an extra charge for rooms facing the water. The hotel manager

wants to survey 120 guests who stayed at the hotel during a convention about

their overall satisfaction with the property.

(a) Explain why choosing a stratified random sample might be preferable to an

SRS in this case. What would you use as strata?

(b) Why might a cluster sample be a simpler option? What would you use as

clusters?

5. Bramlage Coliseum, also known as the Octagon of Doom, has an official capacity

of 12,528. (Although the stadium hosted a record 13,762 fans for the KU game in

1992.)1 The K-State athletic department plans to conduct a survey about

concessions that are sold during games. Tickets are most expensive for seats in

the Loge Box, followed by the chair backs (sections 3-9), and the general

admission seats are the least expensive. A map of the stadium is shown.

(a) The athletic department is considering a stratified random

sample. What would you recommend as the strata? Why?

(b) Explain why a cluster sample might be easier to obtain.

What would you recommend for the clusters? Why?

6. A corporation employs 2000 male and 500 female engineers. A stratified random

sample of 200 male and 50 female engineers gives each engineer a 1/10 chance of

being selected. (Every individual has an equal chance of being selected.)

However, this is not an SRS. Explain why.

1 http://www.kstatesports.com/facilities/bramlage-coliseum.html


Gallup Poll

Look up the article How Polls are Conducted by going to www.gallup.com, clicking the

FAQ link at the bottom of the page, choosing the “How does Gallup polling work?” and

clicking on the link at the end of the answer. Read the article and answer these questions.

1. List three reasons Americans contact the Gallup Organization.

2. What is the fundamental goal of a survey?

3. What is “equal probability of selection” and why is it important?

4. Why does Gallup try to reach people at home?

5. How were the earliest polls conducted? When did this take place?

6. How accurate were exit polls for presidential elections between 1936 and 1984?

7. How did the Gallup polls change in 1986? Why?

8. How did Gallup polls change in 2008? Why?

9. What is the first step in the Gallup process? Explain in detail.


10. What is Gallup's target audience? Who is NOT included?

11. What is the next step in the Gallup process? Explain in detail.

12. Why is it important to go through this “complicated” procedure?

13. Once a household is contacted via landline, how does Gallup choose who to

interview?

14. How many interviews does it take to provide an adequate cross section of Americans?

15. What is more important than the actual number of people who need to be interviewed

for a given sample?

16. Why doesn’t Gallup use samples larger than 1000?

17. What is the “margin of error”?

18. What happens after Gallup collects and processes survey data?


Inference for Sampling (4.1C)

1. Suppose that a sample of households in a community is selected at random

from the telephone directory. Explain how this sampling method results in

under-coverage that could lead to bias.

2. Suppose you want to know the average amount of money spent by fans

attending the opening day for the KC Royals. You get permission to survey at

the stadium, but are forbidden to bother the fans in the club seating or the box

seats (the most expensive seating). Using a computer, you randomly select

500 fans to survey from the rest of the stadium. During the game, you ask

those fans the amount they spent that day.

(a) Why might this lead to a bias result?

(b) Explain why the bias is a result of a sampling error or a nonsampling

error.

3. State whether the following are sampling error or nonsampling error.

Explain.

(a) The subject lies about his drug use.

(b) A typing error is made while recording the data.

(c) Data are gathered by asking people to mail in a survey from the

newspaper.


4. Suppose that a random sample of Manhattan residents were asked, “Recalling

the last ten traffic lights you drove through, how many of them were red when

you entered the intersection?” Of the 880 respondents, 171 admitted that at

least one light had been red.

(a) Can we trust our results? Why or why not?

(b) What is the likely direction of the bias; do you think more or fewer than

171 of the 880 respondents actually ran a red light?

5. Comment on each of the following potential survey questions. Is the question

clear or unclear? Is it unbiased or slanted toward a desired response? Circle

the correct choices.

(a) “Some cell phone users have died while talking on the phone. Do you

support a ban on use of cell phones during driving?”

CLEAR/UNCLEAR UNBIASED/SLANTED

(b) “Do you agree that a national system of health insurance should be

favored because it would provide health care for everyone and would

reduce administrative cost?”


(c) “In view of escalating environmental degradation and incipient resource

depletion, would you favor economic incentives for recycling of resource-

intensive consumer goods?”


(d) “Which of the following best represents your opinion on gun control?

1. The government should confiscate our guns.

2. We have the right to keep and bear arms.”



Observational Studies vs. Experiments (4.2A)

1. A University of Helsinki (Finland) study wanted to determine if chocolate

consumption during pregnancy had an effect on infant temperament at age 6

months. Researchers began asking 305 healthy pregnant women to report their

chocolate consumption. Six months after birth, the researchers asked the women

to report their infants’ temperament, including smiling, laughter and fear. The

babies born to mothers eating chocolate daily during pregnancy were found to be

more active and “positively reactive”- a measure that encompasses smiling and

laughter.

a) Was this an observational study or an experiment? Justify your answer.

b) What are the explanatory and response variables?

c) Does this study show that eating chocolate regularly during pregnancy helps

produce infants with good temperament? Explain.

2. Mrs. Sapp wants to compare the effectiveness of new computer software for

teaching AP Statistics with that of the traditional textbook presentation. She gives

AP Statistics students a pretest to assess their current knowledge base and then

randomly divides them into two groups. One group uses the computer software

and the other uses the textbook. At the end of the year, she tests all the students

again and compares increase in scores between the two groups.

a) Observational study or experiment? Explain.

b) If the computer software group has a much higher average increase in test

scores than the textbook group, what conclusions, if any, could Mrs. Sapp

draw?

3. Binge drinking has been defined as 5 or more drinks for males and 4 or more

drinks for females. An observational study finds that students who binge drink

have lower average GPA than those who don’t. What lurking variables might be

present?


4. An amateur gardener decides to change varieties of tomatoes for this year to see if

the yield is improved. He put in six plants the previous year and six plants this

year using the same part of the garden. The average yield per plant was 11.3

pounds per plant in the previous year and 14.5 pounds per plant using the new

variety. What lurking variables might be present?

5. A mathematics education researcher was interested in determining the effects of

class size (small, medium, or large) and the use of a traditional statistics textbook

versus a new textbook. The researcher conducted her own experiment, assigning

each combination of class size and type of textbook to two classes at each of five

chosen schools. The average final grade for the class was then recorded at the end

of the year.

What are the experimental units?

What are the explanatory variables (factors)? What are the levels?

What are the treatments?

What is the response variable?

6. A student project measured the increase in heart rate of fellow students when they

stepped up and down for three minutes to the beat of a metronome. The step was

either 5 or 10 inches high and the metronome beat was 14, 21, or 28 steps per

minute. Five students stepped at each combination of height and speed.

What are the experimental units?

What are the explanatory variables (factors)? What are the levels?

What are the treatments?

What is the response variable?


Principles of Experimental Design (4.2B)

1. Will cash bonuses motivate teachers to help their students score high on the AP

Statistics Exam? The College Board notes that last year 8% of AP Statistics

teachers had all their students score 3 or higher on the exam. As an experiment,

this year the College Board offers $1000 to teachers whose students all score 3 or

higher on the exam. The percent of teachers whose students score well increases

to 14%. What flaw in the design of this experiment makes it impossible to

determine if the bonus caused the increase?

2. Doctors identify “chronic tension-type headaches” as headaches that occur almost

daily for at least six months. Can antidepressant medication or stress

management training reduce the number and severity of these headaches? Are

both together more effective than alone? Investigators compared four treatments:

antidepressant alone, placebo alone, antidepressant plus stress management,

placebo plus stress management. The headache sufferers named below have

agreed to participate in the study.

Anderson Deng Glaus Nguyen Samuels

Arroyo De Ramos Helling Palmiero Shen

Batista Drasin Husain Percival Suh

Bell Fox Johnson Richards Washburn

Burke Gandhi Martin Rider Zabidi

Cabrera Garcia Murphy

a) Outline the design of the experiment. What is this type of design called?

b) Explain how you would use random assignment to assign the subjects to the

four treatment groups.


3. Coach Schartz hears that a new exercise program will increase upper-body

strength better than lifting weights. He is eager to test his new program in the off-

season with the players on his MHS team. Coach decides to let his players choose

which of the two treatments they will undergo for 3 weeks- exercise or weights.

He will use the number of push-ups a player can do at the end of the experiment

as the response variable.

a) Which principle of experimental design has been violated? Explain how this

could lead to confounding.

b) Comment on the coach’s choice of response variable.

4. A biologist would like to determine which of two brands of weed killer is less

likely to harm the plants at the KSU Gardens. Before spraying near the plants, the

biologist decides to conduct an experiment using 24 individual plants. Which of

the following two plans for randomly assigning the treatments should the

biologist use? Why?

A. Choose the 12 healthiest-looking plants. Apply Brand X week killer to all 12

plants. Apply Brand Y to the remaining 12 plants.

B. Choose 12 of the 24 plants at random. Apply Brand X to those 12 plants.

Apply Brand Y to the remaining 12 plants.

5. Both cluster sampling and stratified random sampling involve viewing the

sampling frame as a collection of subgroups. Explain the difference between

these two types of sampling.


REVIEW:

6. The equation of the least squares regression line for a certain set of data is

ˆ 1.3 0.73y x . What is the residual for the data observation (4, 7)?

7. The heights of adult women are approximately normally distributed about a mean

of 65 inches with a standard deviation of 2 inches. If Rachel is at the 99th

percentile in height for adult women, then her height, in inches, is closest to what

measurement?

8. A company wanted to determine the health care costs of its employees. A sample

of 25 employees were interviewed and their medical expenses for the previous

year were determined. Later the company discovered that the highest medical

expense in the sample was mistakenly recorded as 10 times the actual amount.

However, after correcting the error, the corrected amount was still greater than or

equal to any other medical expense in the sample. Which of the following sample

statistics must have remained the same after the correction was made?

a. Mean b. Median c. Mode d. Range e. Variance

9. Draw the distributions and describe the shape of the data for the following

scenarios:

a) Data set where the mean is greater than the median

b) Data set where the median is greater than the mean

c) Data set where the mean is approximately equal to the median


Experimental Design

For questions 1-3, identify:

a) The experimental units or subjects,

b) The explanatory variable(s) or factor(s) and their levels

c) The treatments

d) The response variable(s)

1. Ability to grow in the shade may help pines found in dry forests of Arizona to resist

drought. How well do these pines grow in shade? Investigators planted pine seedlings in

a greenhouse in either full light, light reduced to 25% of normal by shade cloth, or light

reduced to 5% of normal. At the end of the study, they dried the young trees and

weighed them.

2. You can use your computer to make long-distance telephone calls over the Internet.

How will the cost affect the use of this service? A university plans an experiment to find

out. It will offer the service to all 350 students in one of its dormitories. Some students

will pay a low flat rate. Others will pay higher rates at peak periods and very low rates

off-peak. The university is interested in the amount and time of use and in the effect on

the congestion of the network.

3. Sickle-cell disease is an inherited disorder of the red blood cells that in the US affects

mostly blacks. It can cause severe pain and many complications. Can the drug

hydroxyurea reduce the severe pain caused by sickle-cell disease? A study by the

National Institutes of Health gave the drug to 150 sickle-cell sufferers and a placebo (a

dummy medication) to another 150. The researchers then counted the episodes of pain

reported by each subject.


4. A large study used records from Canada’s national health care system to compare the

effectiveness of two ways to treat prostate disease. The two treatments are traditional

surgery and a new method that does not require surgery. The records described many

patients whose doctors had chosen each method. The study found that patients treated by

the new method were significantly more likely to die within 8 years.

a) Further study of the data showed that this conclusion was wrong. The extra deaths

among patients who got the new method could be explained by lurking variables. What

lurking variables might be confounded with a doctor’s choice of surgical or nonsurgical

treatment?

b) You have 300 prostate patients who are willing to serve as subjects in an experiment

to compare the two methods. Use a diagram to outline the design of a randomized

comparative experiment. Be sure to include the size of the treatment groups and the

response variable.

6. Eight people who suffer from migraine headaches volunteer to take part in an

experiment. Four of these volunteers are to be assigned to receive a new treatment for

migraine headaches and the remaining four are to receive a placebo. Using the numerical

labels attached to the names of the volunteers given above and the list of random digits

given below; randomly assign four of these volunteers to the new treatment. Read the list

of random digits from left to right, starting at the beginning of the list in order to make

your assignment. The names of the volunteers are:

1. Chang 5. Reed

2. Donley 6. Ting

3. Elfring` 7. Toman

4. Miller 8. Whittinghill

05907 50495 11384 44982 20751 27498 12069 45287 71753 90236 66419 84533

(Multiple Choice) The four volunteers assigned to the new treatment are:

A) 0597.

B) Reed, Toman, Reed again, and Miller.

C) Chang, Miller, Reed, and Toman.

D) Reed, Miller, Ting, Donley.


7. Which of the following is not a major principle of experimental design?

A) control.

B) replication.

C) segmentation

D) randomization

8. A study of human development showed two types of movies to groups of children.

Crackers were available in a bowl, and the investigators compared the number of crackers

eaten by children watching the different kinds of movies. One kind of movie was shown

at 8 A.M. (right after the children had breakfast) and another at 11 A.M. (right before the

children had lunch). It was found that during the movie shown at 11 A.M., more crackers

were eaten than during the movie shown at 8 A.M. The investigators concluded that the

different types of movies had an effect on appetite. The results cannot be trusted because

there is confounding. What are some possible lurking variables?

9. A group of college students believes that herbal tea has remarkable restorative powers.

To test their theory they make weekly visits to a local nursing home, visiting with

residents, talking with them, and serving them herbal tea. After several months, many of

the residents are more cheerful and healthy. Can we conclude that herbal tea does

improve one's emotional state, at least for the residents of nursing homes? Why or why

not?

Randomized Block Design (4.2D)

1. Researchers in Japan conducted an experiment on 13 individuals who were

extremely allergic to poison ivy. On one arm, they rubbed a poison ivy leaf and

told the individual it was harmless. On the other arm, each individual was rubbed

with a harmless leaf and told it was poison ivy. All the subjects developed a rash

on the arm where the harmless leaf was rubbed. 11 out of 13 subjects did not

have any reaction to the real poison ivy leaf.

a) What was the placebo in this experiment?

b) Explain how the results of this experiment support the idea of a placebo effect.


2. Happy and Healthy Laboratories has developed a new drug for relieving chronic

pain. 60 patients suffering from arthritis and needing pain relief are available.

Each patient will be treated with either the new drug or a placebo and asked an

hour later, “About what percent of pain relief did you experience?”

a) Why should H&H not simply administer the new drug and record the patients’

responses?

b) Should the patients be told whether they are getting the new drug or the

placebo? How might this knowledge affect their reactions?

c) Suppose that 32 patients receiving the new drug and 18 receiving the placebo

reported significant pain relief. The difference between these two groups is

statistically significant. Explain what this means.

3. Nurse-practitioners often act much like a primary care physician. Are they as

effective at treating chronic conditions? An experiment was conducted with 1316

patients who had been diagnosed with asthma, diabetes, or high blood pressure.

Within each condition, patients were randomly assigned to either a doctor or a

nurse-practitioner. The response variables included measures of the patients’

health and of their satisfaction with their medical care after 6 months.

a) Which are the blocks in this experiment: the medical conditions (asthma, etc.)

or the type of care (nurse or doctor)? Why?

b) Explain why a randomized block design is preferable to a completely

randomized design here.


4. A KSU agriculture researcher wants to compare the yield of 5 corn varieties: A,

B, C, D, and E. The field in which the experiment will be carried out increases in

fertility from north to south. The researcher therefore divides the field into 25

plots of equal size, arranged in 5 east-west rows of 5 plots each, as shown in the

diagram.

a) Explain why a randomized block design would be

better than a completely randomized design in this

setting.

b) Should the researcher use the rows or the columns of the field as blocks?

Explain.

5. Twenty overweight individuals have agreed to participate in a study on the

effectiveness of four weight-loss treatments: A, B, C, and D. The researcher first

calculates how overweight each subject is by comparing her actual weight to her

“ideal” weight. The response variable is the weight loss after 8 weeks of

treatment. Previous studies have shown that the effects of a diet may vary based

on a subject’s initial weight.

a) Explain why a randomized block design would be better than a completely

randomized design in this setting.

b) Suppose the researchers want to form block of size 4. Should the researchers

form the blocks based on the subjects’ last names in alphabetical order or by

how overweight the subjects are? Explain.


6. Recent research suggests that cell phone use while driving distracts the driver.

Manhattan has recently adopted a law that requires hands-free cell phone use by

the driver. However, does hands-free cell phone use also distract the driver?

Suppose our AP Statistics class recruits 40 MHS students for an experiment to

investigate this question. They have a driving simulator equipped with a hands-

free phone for use in the study.

a) We are considering the design shown below. What type of design is it? How

do you know?

b) Explain how blocking could be used to improve the design in (a).

c) Why is it important to randomly assign the treatments within each block?

7. A psychologist wants to know if the difficulty of a task influences our estimate of

how long we spent working at it. She designs two sets of mazes that subjects can

work through on a computer. One set has easy mazes and the other has hard

mazes. Subjects work until told to stop (after 6 minutes, but the subjects do not

know this). They are then asked to estimate how long they worked. There are 30

students available to serve as subjects.

a) Diagram a completely randomized design to learn the effect of difficulty on

the estimated time.

b) Describe a matched pairs experimental design using the same 30 subjects.

c) Which design is more likely to detect a difference in the effects of the

treatments? Explain.

40 MHS

students Random

Assignment

Group 1

20 students

Group 2

20 students

Treatment 1

Drive

Treatment 2

Drive and Talk

Compare

brake time


8. One of the side effects of anti-inflammatory medication used for arthritis pain is

stomach damage (which can lead to ulcers). Researchers are trying to develop an

anti-inflammatory drug that causes minimal stomach damage. An experiment

was designed to test two treatments for arthritis pain (old and new) against a

placebo by measuring their effects on lesions in the stomachs of laboratory rats.

Rats with similar stomach conditions at the start of the experimental period were

randomly assigned to one of the three treatments. Total length of the stomach

lesions (in mm) was measured in each rat after a 2-week period.

a) Experimental units:

Explanatory variables and level(s):

Treatments:

Response Variable:

b) Is this a completely randomized experiment or a randomized blocked one?

Explain.

c) Is the use of a placebo essential here? Why or why not?

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