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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Data set available online
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
Philadelphia
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
Alaska
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
change passwords quarterly?
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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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).
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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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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.