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ACTIVITY 10.1: Comparing the t and z Distributions

# ACTIVITY 10.1: Comparing the t and z Distributions

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Summary of Key Concepts and Formulas

6. Now use the histograms from Step 5 to answer the

following questions:

a. Write a brief description of the shape, center,

and spread for the histogram of the z values. Is

what you see in the histogram consistent with

what you would have expected to see? Explain.

(Hint: In theory, what is the distribution of the

z statistic?)

b. How does the histogram of the t values compare

to the z histogram? Be sure to comment on center, shape, and spread.

c. Is your answer to Part (b) consistent with what

would be expected for a statistic that has a t distribution? Explain.

d. The z and t histograms are based on only 200

samples, and they only approximate the corresponding sampling distributions. The 5th percentile for the standard normal distribution is

507

Ϫ1.645 and the 95th percentile is ϩ1.645. For

a t distribution with df ϭ 5 Ϫ 1 ϭ 4, the 5th

and 95th percentiles are Ϫ2.13 and ϩ2.13, respectively. How do these percentiles compare to

those of the distributions displayed in the histograms? (Hint: Sort the 200 z values—in Minitab,

choose “Sort” from the Manip menu. Once the

values are sorted, percentiles from the histogram

can be found by counting in 10 [which is 5% of

200] values from either end of the sorted list.

Then repeat this with the t values.)

e. Are the results of your simulation and analysis

consistent with the statement that the statistic

x2m

has a standard normal (z) distribuz5

1s/ !n2

x2m

tion and the statistic t 5

has a t distri1s / !n 2

bution? Explain.

A C TI V I T Y 1 0 . 2 A Meaningful Paragraph

Write a meaningful paragraph that includes the following six terms: hypotheses, P-value, reject H0, Type I

error, statistical significance, practical significance.

A “meaningful paragraph” is a coherent piece of

writing in an appropriate context that uses all of the

listed words. The paragraph should show that you un-

derstand the meaning of the terms and their relationship

to one another. A sequence of sentences that just define

the terms is not a meaningful paragraph. When choosing

a context, think carefully about the terms you need to

use. Choosing a good context will make writing a meaningful paragraph easier.

Summary of Key Concepts and Formulas

TERM OR FORMULA

COMMENT

Hypothesis

A claim about the value of a population characteristic.

Null hypothesis, H0

The hypothesis initially assumed to be true. It has the

form H0: population characteristic ϭ hypothesized value.

Alternative hypothesis, Ha

A hypothesis that specifies a claim that is contradictory to

H0 and is judged the more plausible claim when H0 is

rejected.

Type I error

Rejecting H0 when H0 is true; the probability of a Type I

error is denoted by a and is referred to as the significance

level for the test.

Type II error

Not rejecting H0 when H0 is false; the probability of

a Type II error is denoted by b.

Test statistic

A value computed from sample data that is then used as

the basis for making a decision between H0 and Ha.

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508

Chapter 10

Hypothesis Testing Using a Single Sample

TERM OR FORMULA

COMMENT

P-value

The probability, computed assuming H0 to be true, of obtaining a value of the test statistic at least as contradictory

to H0 as what actually resulted. H0 is rejected if P-value Յ

a and not rejected if P-value Ͼ a, where a is the chosen

significance level.

z5

z5

t5

p^ 2 hypothesized value

1hyp. val2 11 2 hyp. val2

n

Å

A test statistic for testing H0: p ϭ hypothesized value

when the sample size is large. The P-value is determined

as an area under the z curve.

x 2 hypothesized value

s

!n

A test statistic for testing H0: m ϭ hypothesized value

when s is known and either the population distribution is

normal or the sample size is large. The P-value is determined as an area under the z curve.

x 2 hypothesized value

s

!n

A test statistic for testing H0: m ϭ hypothesized value

when s is unknown and either the population distribution is normal or the sample size is large. The P-value is

determined from the t curve with df ϭ n Ϫ 1.

Power

The power of a test is the probability of rejecting the null

hypothesis. Power is affected by the size of the difference

between the hypothesized value and the actual value, the

sample size, and the significance level.

Chapter Review Exercises 10.68 - 10.82

10.68 In a representative sample of 1000 adult Americans, only 430 could name at least one justice who is

currently serving on the U.S. Supreme Court (Ipsos,

January 10, 2006). Using a significance level of .01,

carry out a hypothesis test to determine if there is convincing evidence to support the claim that fewer than

half of adult Americans can name at least one justice currently serving on the Supreme Court.

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American

society is a serious problem, and 1283 indicated that

they believe that rudeness is a more serious problem than

in past years (Associated Press, April 3, 2002). Is there

convincing evidence that less than three-quarters of U.S.

adults believe that rudeness is a worsening problem? Test

the relevant hypotheses using a significance level of .05.

10.69

10.70 Students at the Akademia Podlaka conducted an

experiment to determine whether the Belgium-minted

Euro coin was equally likely to land heads up or tails up.

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Coins were spun on a smooth surface, and in 250 spins,

140 landed with the heads side up (New Scientist, January 4, 2002). Should the students interpret this result as

convincing evidence that the proportion of the time the

coin would land heads up is not .5? Test the relevant

hypotheses using a ϭ .01. Would your conclusion be

different if a significance level of .05 had been used?

Explain.

10.71 An article titled “Teen Boys Forget Whatever It

Was” appeared in the Australian newspaper The Mercury

(April 21, 1997). It described a study of academic performance and attention span and reported that the mean

time to distraction for teenage boys working on an independent task was 4 minutes. Although the sample size

was not given in the article, suppose that this mean was

based on a random sample of 50 teenage Australian boys

and that the sample standard deviation was 1.4 minutes.

Is there convincing evidence that the average attention

span for teenage boys is less than 5 minutes? Test the

relevant hypotheses using a ϭ .01.

Video Solution available

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Chapter Review Exercises

10.72 The authors of the article “Perceived Risks of

Heart Disease and Cancer Among Cigarette Smokers”

(Journal of the American Medical Association [1999]:

1019–1021) expressed the concern that a majority of smokers do not view themselves as being at increased risk of

heart disease or cancer. A study of 737 current smokers

selected at random from U.S. households with telephones

found that of the 737 smokers surveyed, 295 indicated

that they believed they have a higher than average risk of

cancer. Do these data suggest that p, the true proportion

of smokers who view themselves as being at increased risk

of cancer is in fact less than .5, as claimed by the authors

of the paper? Test the relevant hypotheses using a ϭ .05.

10.73 A number of initiatives on the topic of legalized

gambling have appeared on state ballots. Suppose that a

political candidate has decided to support legalization of

casino gambling if he is convinced that more than twothirds of U.S. adults approve of casino gambling. Suppose

that 1523 adults (selected at random from households

with telephones) were asked whether they approved of

casino gambling. The number in the sample who approved was 1035. Does the sample provide convincing

evidence that more than two-thirds approve?

10.74 Although arsenic is known to be a poison, it also

has some beneficial medicinal uses. In one study of the

use of arsenic to treat acute promyelocytic leukemia

(APL), a rare type of blood cell cancer, APL patients

were given an arsenic compound as part of their treatment. Of those receiving arsenic, 42% were in remission

and showed no signs of leukemia in a subsequent examination (Washington Post, November 5, 1998). It is

known that 15% of APL patients go into remission after

the conventional treatment. Suppose that the study had

included 100 randomly selected patients (the actual

number in the study was much smaller). Is there sufficient evidence to conclude that the proportion in remission for the arsenic treatment is greater than .15, the remission proportion for the conventional treatment? Test

the relevant hypotheses using a .01 significance level.

10.75 Many people have misconceptions about how

profitable small, consistent investments can be. In a survey of 1010 randomly selected U.S. adults (Associated

Press, October 29, 1999), only 374 responded that they

thought that an investment of \$25 per week over 40 years

with a 7% annual return would result in a sum of over

\$100,000 (the correct amount is \$286,640). Is there sufficient evidence to conclude that less than 40% of U.S.

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Data set available online

509

adults are aware that such an investment would result in

a sum of over \$100,000? Test the relevant hypotheses

using ␣ ϭ .05.

10.76 The same survey described in the previous exercise

also asked the individuals in the sample what they thought

was their best chance to obtain more than \$500,000 in

their lifetime. Twenty-eight percent responded “win a lottery or sweepstakes.” Does this provide convincing evidence that more than one-fourth of U.S. adults see a lottery or sweepstakes win as their best chance of accumulating

\$500,000? Carry out a test using a significance level of .01.

10.77 Speed, size, and strength are thought to be important factors in football performance. The article

“Physical and Performance Characteristics of NCAA

Division I Football Players” (Research Quarterly for

Exercise and Sport [1990]: 395–401) reported on physical characteristics of Division I starting football players

in the 1988 football season. Information for teams

ranked in the top 20 was easily obtained, and it was reported that the mean weight of starters on top-20 teams

was 105 kg. A random sample of 33 starting players

(various positions were represented) from Division I

teams that were not ranked in the top 20 resulted in a

sample mean weight of 103.3 kg and a sample standard

deviation of 16.3 kg. Is there sufficient evidence to conclude that the mean weight for non-top-20 starters is less

than 105, the known value for top-20 teams?

10.78 Duck hunting in populated areas faces opposition on the basis of safety and environmental issues. In a

survey to assess public opinion regarding duck hunting

on Morro Bay (located along the central coast of California), a random sample of 750 local residents included

560 who strongly opposed hunting on the bay. Does this

sample provide sufficient evidence to conclude that the

majority of local residents oppose hunting on Morro

Bay? Test the relevant hypotheses using a ϭ .01.

10.79 Past experience has indicated that the true response rate is 40% when individuals are approached with

a request to fill out and return a particular questionnaire

in a stamped and addressed envelope. An investigator

believes that if the person distributing the questionnaire

is stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend

to respond at a rate higher than 40%. To investigate this

theory, a distributor is fitted with an eye patch. Of the

200 questionnaires distributed by this individual, 109

were returned. Does this strongly suggest that the reVideo Solution available

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510

Chapter 10

Hypothesis Testing Using a Single Sample

sponse rate in this situation exceeds the rate in the past?

State and test the appropriate hypotheses at significance

level .05.

10.80

An automobile manufacturer who wishes to

advertise that one of its models achieves 30 mpg (miles

per gallon) decides to carry out a fuel efficiency test. Six

nonprofessional drivers are selected, and each one drives

a car from Phoenix to Los Angeles. The resulting fuel

efficiencies (in miles per gallon) are:

27.2 29.3 31.2 28.4 30.3 29.6

Assuming that fuel efficiency is normally distributed under these circumstances, do the data contradict the claim

that true average fuel efficiency is (at least) 30 mpg?

10.81 A student organization uses the proceeds from a

particular soft-drink dispensing machine to finance its

activities. The price per can had been \$0.75 for a long

time, and the average daily revenue during that period

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Data set available online

had been \$75.00. The price was recently increased to

\$1.00 per can. A random sample of n ϭ 20 days after the

price increase yielded a sample mean daily revenue and

sample standard deviation of \$70.00 and \$4.20, respectively. Does this information suggest that the true average daily revenue has decreased from its value before the

price increase? Test the appropriate hypotheses using

a ϭ .05.

10.82 A hot tub manufacturer advertises that with its

heating equipment, a temperature of 100ЊF can be

achieved on average in 15 minutes or less. A random

sample of 25 tubs is selected, and the time necessary to

achieve a 100ЊF temperature is determined for each tub.

The sample mean time and sample standard deviation

are 17.5 minutes and 2.2 minutes, respectively. Does this

information cast doubt on the company’s claim? Carry

out a test of hypotheses using significance level .05.

Video Solution available

Cumulative Review Exercises CR10.1 - CR10.16

CR10.2 The following graphical display appeared in

USA Today (June 3, 2009). Write a few sentences cri-

USA TODAY Snapshots®

Report card: Roads getting an ‘F’

States with the highest and lowest percentage of

roads in “poor” condition:

Source: American Association of State Highway

and Transportation Officials

tiquing this graphical display. Do you think it does a

good job of creating a visual representation of the three

percentages in the display?

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Data set available online

By Anne R. Carey and Dave Merrill, USA TODAY

USA TODAY. June 3, 2009. Reprinted with permission.

CR10.1 The AARP Bulletin (March 2010) included the

following short news brief:“Older adults who did 1 hour

of tai chi twice weekly cut their pain from knee osteoarthritis considerably in a 12-week study conducted at

Tufts University School of Medicine.” Suppose you were

asked to design a study to investigate this claim. Describe

an experiment that would allow comparison of the reduction in knee pain for those who did 1 hour of tai chi

twice weekly to the reduction in knee pain for those who

did not do tai chi. Include a discussion of how study

participants would be selected, how pain reduction

would be measured, and how participants would be assigned to experimental groups.

Video Solution available

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Cumulative Review Exercises

CR10.3

The article “Flyers Trapped on Tarmac

Average Appointment

Wait Time

Push for Rules on Release” (USA Today, July 28, 2009)

City

included the accompanying data on the number of

flights with a tarmac delay of more than 3 hours between

October 2008 and May 2009 for U.S. airlines.

Atlanta

Boston

11.2

49.6

Dallas

Denver

Detroit

Houston

Los Angeles

Miami

Minneapolis

New York

Portland

San Diego

Seattle

Washington, D.C.

19.2

15.4

12.0

23.4

24.2

15.4

19.8

19.2

27.0

14.4

20.2

14.2

22.6

Airline

Number of

Flights

Rate per 100,000

Flights

7

0

48

44

11

29

72

81

93

5

0

18

17

24

13

29

11

29

46

0.4

0.0

1.3

1.6

0.6

2.7

4.1

2.8

4.9

0.9

0.0

1.4

1.1

1.2

0.7

0.8

0.1

1.1

1.6

AirTran

American

American Eagle

Atlantic Southeast

Comair

Continental

Delta

ExpressJet

Frontier

Hawaiian

JetBlue

Mesa

Northwest

Pinnacle

SkyWest

Southwest

United

US Airways

a. Construct a dotplot of the data on number of flights

delayed for more than 3 hours. Are there any unusual observations that stand out in the dot plot?

What airlines appear to be the worst in terms of

number of flights delayed on the tarmac for more

than 3 hours?

b. Construct a dotplot of the data on rate per 100,000

flights. Write a few sentences describing the interesting features of this plot.

c. If you wanted to compare airlines on the basis of

tarmac delays, would you recommend using the data

on number of flights delayed or on rate per 100,000

flights? Explain the reason for your choice.

The article “Wait Times on Rise to See Doctor” (USA Today, June 4, 2009) gave the accompanying

data on average wait times in days to get an appointment

with a medical specialist in 15 U.S. cities. Construct a

boxplot of the average wait-time data. Are there any outliers in the data set?

CR10.4

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Data set available online

511

CR10.5 The report “New Study Shows Need for

Americans to Focus on Securing Online Accounts and

Backing up Critical Data” (PRNewswire, October 29,

2009) reported that only 25% of Americans change

computer passwords quarterly, in spite of a recommendation from the National Cyber Security Alliance that

passwords be changed at least once every 90 days. For

purposes of this exercise, assume that the 25% figure is

correct for the population of adult Americans.

a. If a random sample of 20 adult Americans is selected, what is the probability that exactly 3 of them

b. What is the probability that more than 8 people in a

random sample of 20 adult Americans change passwords quarterly?

c. What is the mean and standard deviation of the variable x ϭ number of people in a random sample of

100 adult Americans who change passwords quarterly?

d. Find the approximate probability that the number of

people who change passwords quarterly in a random

sample of 100 adult Americans is less than 20.

CR10.6 The article “Should Canada Allow Direct-toConsumer Advertising of Prescription Drugs?” (Canadian Family Physician [2009]: 130–131) calls for the legalization of advertising of prescription drugs in Canada.

Suppose you wanted to conduct a survey to estimate the

proportion of Canadians who would support allowing

this type of advertising. How large a random sample

would be required to estimate this proportion to within

.02 with 95% confidence?

Video Solution available

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Chapter 10

Hypothesis Testing Using a Single Sample

CR10.7 The National Association of Colleges and

Employers carries out a student survey each year. A summary of data from the 2009 survey included the following information:

• 26% of students graduating in 2009 intended to go

on to graduate or professional school.

• Only 40% of those who graduated in 2009 received

at least one job offer prior to graduation.

• Of those who received a job offer, only 45% had

accepted an offer by the time they graduated.

Consider the following events:

O ϭ event that a randomly selected 2009 graduate

received at least one job offer

A ϭ event that a randomly selected 2009 graduate

accepted a job offer prior to graduation

G ϭ event that a randomly selected 2009 graduate

plans to attend graduate or professional school

Compute the following probabilities.

a. P 1O2

b. P 1A2

c. P 1G2

d. P 1A 0 O2

e. P 1O 0 A2

f. P 1A ʝ O2

CR10.8 It probably wouldn’t surprise you to know that

Valentine’s Day means big business for florists, jewelry

stores, and restaurants. But would it surprise you to

know that it is also a big day for pet stores? In January

2010, the National Retail Federation conducted a survey

of consumers who they believed were selected in a way

that would produce a sample representative of the population of adults in the United States (“This Valentine’s

Day, Couples Cut Back on Gifts to Each Other, According to NRF Survey,” www.nrf.com). One of the

questions in the survey asked if the respondent planned

to spend money on a Valentine’s Day gift for his or her

pet this year.

a. The proportion who responded that they did plan to

purchase a gift for their pet was .173. Suppose that

the sample size for this survey was n ϭ 200. Construct and interpret a 95% confidence interval for

the proportion of all U.S. adults who planned to

purchase a Valentine’s Day gift for their pet in 2010.

b. The actual sample size for the survey was much

larger than 200. Would a 95% confidence interval

computed using the actual sample size have been

narrower or wider than the confidence interval computed in Part (a)?

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Data set available online

c. Still assuming a sample size of n ϭ 200, carry out a

hypothesis test to determine if the data provides

convincing evidence that the proportion who

planned to buy a Valentine’s Day gift for their pet in

2010 was greater than .15. Use a significance level

of .05.

CR10.9 The article “Doctors Cite Burnout in Mistakes” (San Luis Obispo Tribune, March 5, 2002) reported that many doctors who are completing their residency have financial struggles that could interfere with

training. In a sample of 115 residents, 38 reported that

they worked moonlighting jobs and 22 reported a credit

card debt of more than \$3000. Suppose that it is reasonable to consider this sample of 115 as a random sample

of all medical residents in the United States.

a. Construct and interpret a 95% confidence interval

for the proportion of U.S. medical residents who

work moonlighting jobs.

b. Construct and interpret a 90% confidence interval

for the proportion of U.S. medical residents who

have a credit card debt of more than \$3000.

c. Give two reasons why the confidence interval in Part

(a) is wider than the confidence interval in Part (b).

CR10.10 The National Geographic Society conducted

a study that included 3000 respondents, age 18 to 24,

in nine different countries (San Luis Obispo Tribune,

November 21, 2002). The society found that 10% of the

participants could not identify their own country on a

blank world map.

a. Construct a 90% confidence interval for the proportion who can identify their own country on a blank

world map.

b. What assumptions are necessary for the confidence

interval in Part (a) to be valid?

c. To what population would it be reasonable to generalize the confidence interval estimate from Part (a)?

CR10.11 “Heinz Plays Catch-up After Under-Filling

Ketchup Containers” is the headline of an article that

appeared on CNN.com (November 30, 2000). The

article stated that Heinz had agreed to put an extra 1%

of ketchup into each ketchup container sold in California for a 1-year period. Suppose that you want to make

sure that Heinz is in fact fulfilling its end of the agreement. You plan to take a sample of 20-oz bottles shipped

to California, measure the amount of ketchup in each

bottle, and then use the resulting data to estimate the

mean amount of ketchup in each bottle. A small pilot

Video Solution available

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Cumulative Review Exercises

513

study showed that the amount of ketchup in 20-oz bottles varied from 19.9 to 20.3 oz. How many bottles

should be included in the sample if you want to estimate

the true mean amount of ketchup to within 0.1 oz with

95% confidence?

community. Suppose that this result was based on a

sample of 512 religion surfers. Is there convincing evidence that the proportion of religion surfers who belong

to a religious community is different from .68, the proportion for the general population? Use a ϭ .05.

CR10.12 In a survey conducted by Yahoo Small Busi-

CR10.15 A survey of teenagers and parents in Canada

ness, 1432 of 1813 adults surveyed said that they would

alter their shopping habits if gas prices remain high (Associated Press, November 30, 2005). The article did

not say how the sample was selected, but for purposes of

this exercise, assume that it is reasonable to regard this

sample as representative of adult Americans. Based on

these survey data, is it reasonable to conclude that more

than three-quarters of adult Americans plan to alter their

shopping habits if gas prices remain high?

conducted by the polling organization Ipsos (“Untangling

CR10.13 In an AP-AOL sports poll (Associated Press,

December 18, 2005), 272 of 394 randomly selected

baseball fans stated that they thought the designated hitter rule should either be expanded to both baseball

leagues or eliminated. Based on the given information, is

there sufficient evidence to conclude that a majority of

baseball fans feel this way?

The article “Americans Seek Spiritual Guidance on Web” (San Luis Obispo Tribune, October 12,

2002) reported that 68% of the general population be-

CR10.14

long to a religious community. In a survey on Internet

use, 84% of “religion surfers” (defined as those who seek

spiritual help online or who have used the web to search

for prayer and devotional resources) belong to a religious

Bold exercises answered in back

Data set available online

the Web: The Facts About Kids and the Internet,” January 25, 2006) included questions about Internet use. It

was reported that for a sample of 534 randomly selected

teens, the mean number of hours per week spent online

was 14.6 and the standard deviation was 11.6.

a. What does the large standard deviation, 11.6 hours,

tell you about the distribution of online times for

this sample of teens?

b. Do the sample data provide convincing evidence

that the mean number of hours that teens spend

online is greater than 10 hours per week?

CR10.16 The same survey referenced in the previous

exercise reported that for a random sample of 676 parents of Canadian teens, the mean number of hours parents thought their teens spent online was 6.5 and the

sample standard deviation was 8.6.

a. Do the sample data provide convincing evidence

that the mean number of hours that parents think

their teens spend online is less than 10 hours per

week?

b. Write a few sentences commenting on the results of

the test in Part (a) and of the test in Part (b) of the

previous exercise.

Video Solution available

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CHAPTER

11

Comparing

Two Populations

or Treatments

Many investigations are carried out for the purpose of

comparing two populations or treatments. For example,

the article “What Do Happy People Do?” (Social Indicators Research [2008]: 565–571) investigates differences in

the way happy people and unhappy people spend their

time. By comparing data from a large national sample of

people who described themselves as very happy to data

from a large national sample of people who described

themselves as not happy, the authors were able to investigate whether the mean amount of time spent in various

activities was higher for one group than for the other. Using hypothesis tests to be introduced in this chapter, the

Andersen Ross/Digital Vision/Jupiter Images

authors were able to conclude that there was no significant difference in the mean number of hours per day spent on the Internet for happy and

unhappy people but that the mean number of hours per day spent watching TV was

significantly higher for unhappy people. In this chapter, we will see hypothesis tests

and confidence intervals that can be used to compare two populations or treatments.

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515

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516

Chapter 11

Comparing Two Populations or Treatments

Inferences Concerning the Difference Between

Two Population or Treatment Means Using

Independent Samples

11.1

In this section, we consider using sample data to compare two population means or

two treatment means. An investigator may wish to estimate the difference between

two population means or to test hypotheses about this difference. For example, a

university financial aid director may want to determine whether the mean cost of

textbooks is different for students enrolled in the engineering college than for students enrolled in the liberal arts college. Here, two populations (one consisting of all

students enrolled in the engineering college and the other consisting of all students

enrolled in the liberal arts college) are to be compared on the basis of their respective

mean textbook costs. Information from two random samples, one from each population, could be the basis for making such a comparison.

In other cases, an experiment might be carried out to compare two different treatments or to compare the effect of a treatment with the effect of no treatment. For

example, an agricultural experimenter might wish to compare weight gains for animals placed on two different diets (each diet is a treatment), or an educational researcher might wish to compare online instruction to traditional classroom instruction by studying the difference in mean scores on a common final exam (each type of

instruction is a treatment).

In previous chapters, the symbol m was used to denote the mean of a single population under study. When comparing two populations or treatments, we must use

notation that distinguishes between the characteristics of the first and those of the

second. This is accomplished by using subscripts, as shown in the accompanying box.

Notation

Mean

Variance

Standard

Deviation

m1

m2

s21

s22

s1

s2

Population or Treatment 1

Population or Treatment 2

Sample from Population or Treatment 1

Sample from Population or Treatment 2

Sample

Size

Mean

Variance

Standard

Deviation

n1

n2

x1

x2

s 21

s 22

s1

s2

A comparison of means focuses on the difference, m1 Ϫ m2. When m1 Ϫ m2 ϭ 0,

the two population or treatment means are identical. That is,

m1 Ϫ m25 0 is equivalent to m1 ϭ m2

Similarly,

m1 Ϫ m2 Ͼ 0 is equivalent to m1 Ͼ m2

and

m1 Ϫ m2 Ͻ 0 is equivalent to m1 Ͻ m2

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11.1

Inferences Concerning the Difference Between Two Population or Treatment Means Using Independent Samples

517

Before developing inferential procedures concerning m1 Ϫ m2, we must consider

how the two samples, one from each population, are selected. Two samples are said

to be independent samples if the selection of the individuals or objects that make up

one sample does not influence the selection of individuals or objects in the other

sample. However, when observations from the first sample are paired in some meaningful way with observations in the second sample, the samples are said to be paired.

For example, to study the effectiveness of a speed-reading course, the reading speed

of subjects could be measured before they take the class and again after they complete

the course. This gives rise to two related samples—one from the population of individuals who have not taken this particular course (the “before” measurements) and

one from the population of individuals who have had such a course (the “after” measurements). These samples are paired. The two samples are not independently chosen, because the selection of individuals from the first (before) population completely

determines which individuals make up the sample from the second (after) population.

In this section, we consider procedures based on independent samples. Methods for

analyzing data resulting from paired samples are presented in Section 11.2.

Because x1 provides an estimate of m1 and x2 gives an estimate of m2, it is natural

to use x1 2 x2 as a point estimate of m1 Ϫ m2. The value of x1 varies from sample to

sample (it is a statistic), as does the value of x2. Since the difference x1 2 x2 is calculated from sample values, it is also a statistic and, therefore, has a sampling

distribution.

Properties of the Sampling Distribution of x1 2 x2

If the random samples on which x1 and x2 are based are selected independently

of one another, then

1. mx1 2x2 5 a

mean value

b 5 m x1 2 m x2 5 m 1 2 m 2

of x1 2 x2

The sampling distribution of x1 2 x2 is always centered at the value of

m1 2 m2, so x1 2 x2 is an unbiased statistic for estimating m1 2 m2.

2. s2x12x2 5 a

and

variance of

s21

s22

2

2

5

s

1

s

5

1

b

x1

x2

n1

n2

x1 2 x2

sx1 2x2 5 a

x1 2 x2

standard deviation

s21

s22

1

b5

n2

of x1 2 x2

Å n1

3. If n1 and n2 are both large or the population distributions are (at least

approximately) normal, x1 and x2 each have (at least approximately) a normal distribution. This implies that the sampling distribution of x1 2 x2 is

also normal or approximately normal.

Properties 1 and 2 follow from the following general results:

1. The mean value of a difference in means is the difference of the two individual

mean values.

2. The variance of a difference of independent quantities is the sum of the two individual variances.

When the sample sizes are large or when the population distributions are approximately normal, the properties of the sampling distribution of x1 2 x2 imply that

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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