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**INFERENCES FOR THE DIFFERENCE BETWEEN **

TWO PROPORTIONS ON THE COMPUTER
It is so common to test against the null hypothesis of no difference between the two true proportions that moststatistics programs simply assume this null hypothesis. And most will automatically use the pooled standarddeviation. If you wish to test a different null (say, that the true difference is 0.3), you may have to search for away to do it.

Many statistics packages don't offer special commands for inference for differences between proportions. Aswith inference for single proportions, most statistics programs want the "success" and "failure" status for eachcase. Usually these are given as 1 or 0, but they might be category names like "yes" and "no." Often we just knowthe proportions of successes,

*p*N1 and

*p*N2, and the counts,

*n*1 and

*n*2. Computer packages don't usually deal withsummary data like these easily. Calculators typically do a better job.

**1. Online social networking. **The Parents & Teens 2006

*z*-test could tell whether there was a real difference
Survey of 935 12- to 17-year-olds found that, among teens
among all teens. Explain why that test would not be ap-
aged 15–17, girls were significantly more likely to have
propriate for these data.

used social networking sites and online profiles. 70% of

**6. Regulating access. **When a random sample of 935 par-

the girls surveyed had used an online social network,
ents were asked about rules in their homes, 77% said they
compared to 54% of the boys. What does it mean to say
had rules about the kinds of TV shows their children
that the difference in proportions is "significant"?
could watch. Among the 790 of those parents whose

**2. Science news. **In 2007 a Pew survey asked 1447 Inter-

teenage children had Internet access, 85% had rules about
net users about their sources of news and information
the kinds of Internet sites their teens could visit. That
about science. Among those who had broadband access
looks like a difference, but can we tell? Explain why a
at home, 34% said they would turn to the Internet for
two-sample

*z*-test would not be appropriate here.

most of their science news. The report on this survey

**7. Gender gap.**
A presidential candidate fears he has a
claims that this is not significantly different from the per-
problem with women voters. His campaign staff plans
centage (33%) who said they ordinarily get their science
to run a poll to assess the situation. They'll randomly
news from television. What does it mean to say that the
sample 300 men and 300 women, asking if they have a
difference is not significant?
favorable impression of the candidate. Obviously, the

**3. Name recognition. **A political candidate runs a week-

staff can't know this, but suppose the candidate has a
long series of TV ads designed to attract public attention
positive image with 59% of males but with only 53% of
to his campaign. Polls taken before and after the ad cam-
paign show some increase in the proportion of voters
a) What sampling design is his staff planning to use?
who now recognize this candidate's name, with a
b) What difference would you expect the poll to show?
P-value of 0.033. Is it reasonable to believe the ads may
c) Of course, sampling error means the poll won't reflect
the difference perfectly. What's the standard deviationfor the difference in the proportions?

**4. Origins. **In a 1993 Gallup poll, 47% of the respondents

d) Sketch a sampling model for the size difference in
agreed with the statement "

*God created human beings pretty*
proportions of men and women with favorable im-

*much in their present form at one time within the last 10,000*
pressions of this candidate that might appear in a poll

*years or so*." When Gallup asked the same question in
2001, only 45% of those respondents agreed. Is it reason-
e) Could the campaign be misled by the poll, concluding
able to conclude that there was a change in public opin-
that there really is no gender gap? Explain.

ion given that the P-value is 0.37? Explain.

**8. Buy it again? **A consumer magazine plans to poll car

**5. Revealing information. **886 randomly sampled teens

owners to see if they are happy enough with their vehi-
were asked which of several personal items of informa-
cles that they would purchase the same model again.

tion they thought it okay to share with someone they had
They'll randomly select 450 owners of American-made
just met. 44% said it was okay to share their e-mail ad-
cars and 450 owners of Japanese models. Obviously, the
dresses, but only 29% said they would give out their cell
actual opinions of the entire population couldn't be
phone numbers. A researcher claims that a two-proportion
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CHAPTER 22 Comparing Two Proportions
known, but suppose 76% of owners of American cars and

**13. Ear infections. **A new vaccine was recently tested to

78% of owners of Japanese cars would purchase another.

see if it could prevent the painful and recurrent ear infec-
a) What sampling design is the magazine planning to use?
tions that many infants suffer from.

*The Lancet*, a medical
b) What difference would you expect their poll to show?
journal, reported a study in which babies about a year
c) Of course, sampling error means the poll won't reflect
old were randomly divided into two groups. One group
the difference perfectly. What's the standard deviation
received vaccinations; the other did not. During the fol-
for the difference in the proportions?
lowing year, only 333 of 2455 vaccinated children had ear
d) Sketch a sampling model for the difference in propor-
infections, compared to 499 of 2452 unvaccinated children
tions that might appear in a poll like this.

in the control group.

e) Could the magazine be misled by the poll, concluding
a) Are the conditions for inference satisfied?
that owners of American cars are much happier with
b) Find a 95% confidence interval for the difference in
their vehicles than owners of Japanese cars? Explain.

rates of ear infection.

**9. Arthritis. **The Centers for Disease Control and Preven-

c) Use your confidence interval to explain whether you
tion reported a survey of randomly selected Americans
think the vaccine is effective.

age 65 and older, which found that 411 of 1012 men and

**14. Anorexia. **The

*Journal of the American Medical Associa-*
535 of 1062 women suffered from some form of arthritis.

*tion *reported on an experiment intended to see if the drug
a) Are the assumptions and conditions necessary for in-
Prozac® could be used as a treatment for the eating disor-
ference satisfied? Explain.

der anorexia nervosa. The subjects, women being treated
b) Create a 95% confidence interval for the difference in
for anorexia, were randomly divided into two groups. Of
the proportions of senior men and women who have
the 49 who received Prozac, 35 were deemed healthy a
this disease.

year later, compared to 32 of the 44 who got the placebo.

c) Interpret your interval in this context.

a) Are the conditions for inference satisfied?
d) Does this confidence interval suggest that arthritis is
b) Find a 95% confidence interval for the difference in
more likely to afflict women than men? Explain.

**10. Graduation. **In October 2000 the U.S. Department of

c) Use your confidence interval to explain whether you
Commerce reported the results of a large-scale survey on
think Prozac is effective.

high school graduation. Researchers contacted more than

**15. Another ear infection. **In Exercise 13 you used a con-

25,000 Americans aged 24 years to see if they had fin-
fidence interval to examine the effectiveness of a vaccine
ished high school; 84.9% of the 12,460 males and 88.1% of
against ear infections in babies. Suppose that instead you
the 12,678 females indicated that they had high school
had conducted a hypothesis test. (Answer these questions

*without *actually doing the test.)
a) Are the assumptions and conditions necessary for in-
a) What hypotheses would you test?
ference satisfied? Explain.

b) State a conclusion based on your confidence interval.

b) Create a 95% confidence interval for the difference in
c) What alpha level did your test use?
graduation rates between males and females.

d) If that conclusion is wrong, which type of error did
c) Interpret your confidence interval.

d) Does this provide strong evidence that girls are more
e) What would be the consequences of such an error?
likely than boys to complete high school? Explain.

**16. Anorexia again. **In Exercise 14 you used a confidence

**11. Pets. **Researchers at the National Cancer Institute re-

interval to examine the effectiveness of Prozac in treating
leased the results of a study that investigated the effect of
anorexia nervosa. Suppose that instead you had conducted
weed-killing herbicides on house pets. They examined
a hypothesis test. (Answer these questions

*without *actu-
827 dogs from homes where an herbicide was used on a
ally doing the test.)
regular basis, diagnosing malignant lymphoma in 473 of
a) What hypotheses would you test?
them. Of the 130 dogs from homes where no herbicides
b) State a conclusion based on your confidence
were used, only 19 were found to have lymphoma.

a) What's the standard error of the difference in the two
c) What alpha level did your test use?
d) If that conclusion is wrong, which type of error did
b) Construct a 95% confidence interval for this difference.

c) State an appropriate conclusion.

e) What would be the consequences of such an error?

**12. Carpal tunnel. **The painful wrist condition called

**17. Teen smoking, part I. **A Vermont study published in

carpal tunnel syndrome can be treated with surgery or
December 2001 by the American Academy of Pediatrics
less invasive wrist splints. In September 2002,

*Time *maga-
examined parental influence on teenagers' decisions to
zine reported on a study of 176 patients. Among the half
smoke. A group of students who had never smoked were
that had surgery, 80% showed improvement after three
questioned about their parents' attitudes toward smok-
months, but only 54% of those who used the wrist splints
ing. These students were questioned again two years later
to see if they had started smoking. The researchers found
a) What's the standard error of the difference in the two
that, among the 284 students who indicated that their
parents disapproved of kids smoking, 54 had become es-
b) Construct a 95% confidence interval for this difference.

tablished smokers. Among the 41 students who initially
c) State an appropriate conclusion.

said their parents were lenient about smoking, 11 became
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smokers. Do these data provide strong evidence that
Researchers found that 8% of 182 babies born to mothers
parental attitude influences teenagers' decisions about
who were exposed to heavy doses of soot and ash on Sep-
tember 11 were classified as having low birth weight.

a) What kind of design did the researchers use?
Only 4% of 2300 babies born in another New York City
b) Write appropriate hypotheses.

hospital whose mothers had not been near the site of the
c) Are the assumptions and conditions necessary for
disaster were similarly classified. Does this indicate a
inference satisfied?
possibility that air pollution might be linked to a signifi-
d) Test the hypothesis and state your conclusion.

cantly higher proportion of low-weight babies?
e) Explain in this context what your P-value means.

a) Was this an experiment? Explain.

f) If that conclusion is actually wrong, which type of
b) Test an appropriate hypothesis and state your conclu-
error did you commit?
sion in context.

**18. Depression. **A study published in the

*Archives of Gen-*
c) If you concluded there is a difference, estimate that

*eral Psychiatry *in March 2001 examined the impact of de-
difference with a confidence interval and interpret
pression on a patient's ability to survive cardiac disease.

that interval in context.

Researchers identified 450 people with cardiac disease,

**23. Politics and sex. **One month before the election, a poll

evaluated them for depression, and followed the group
of 630 randomly selected voters showed 54% planning
for 4 years. Of the 361 patients with no depression,
to vote for a certain candidate. A week later it became
67 died. Of the 89 patients with minor or major depres-
known that he had had an extramarital affair, and a new
sion, 26 died. Among people who suffer from cardiac
poll showed only 51% of 1010 voters supporting him. Do
disease, are depressed patients more likely to die than
these results indicate a decrease in voter support for his
non-depressed ones?
a) What kind of design was used to collect these data?
a) Test an appropriate hypothesis and state your
b) Write appropriate hypotheses.

c) Are the assumptions and conditions necessary for in-
b) If your conclusion turns out to be wrong, did you
ference satisfied?
make a Type I or Type II error?
d) Test the hypothesis and state your conclusion.

c) If you concluded there was a difference, estimate that
e) Explain in this context what your P-value means.

difference with a confidence interval and interpret
f) If your conclusion is actually incorrect, which type of
your interval in context.

error did you commit?

**24. Shopping. **A survey of 430 randomly chosen adults

**19. Teen smoking, part II. **Consider again the Vermont

found that 21% of the 222 men and 18% of the 208 women
study discussed in Exercise 17.

had purchased books online.

a) Create a 95% confidence interval for the difference in
a) Is there evidence that men are more likely than
the proportion of children who may smoke and have
women to make online purchases of books? Test an
approving parents and those who may smoke and
appropriate hypothesis and state your conclusion in
have disapproving parents.

b) Interpret your interval in this context.

b) If your conclusion in fact proves to be wrong, did you
c) Carefully explain what "95% confidence" means.

make a Type I or Type II error?

**20. Depression revisited. **Consider again the study of the

c) Estimate this difference with a confidence interval.

association between depression and cardiac disease sur-
d) Interpret your interval in context.

vivability in Exercise 18.

**25. Twins. **In 2001, one county reported that, among 3132

a) Create a 95% confidence interval for the difference in
white women who had babies, 94 were multiple births.

survival rates.

There were also 20 multiple births to 606 black women.

b) Interpret your interval in this context.

Does this indicate any racial difference in the likelihood
c) Carefully explain what "95% confidence" means.

of multiple births?

**21. Pregnancy. **In 1998, a San Diego reproductive clinic re-

a) Test an appropriate hypothesis and state your conclu-
ported 42 live births to 157 women under the age of 38,
sion in context.

but only 7 live births for 89 clients aged 38 and older. Is
b) If your conclusion is incorrect, which type of error did
this strong evidence of a difference in the effectiveness of
the clinic's methods for older women?

**26. Mammograms. **A 9-year study in Sweden compared

a) Was this an experiment? Explain.

21,088 women who had mammograms with 21,195 who
b) Test an appropriate hypothesis and state your conclu-
did not. Of the women who underwent screening,
sion in context.

63 died of breast cancer, compared to 66 deaths among
c) If you concluded there was a difference, estimate that
the control group. (

*The New York Times*, Dec 9, 2001)
difference with a confidence interval and interpret
a) Do these results support the effectiveness of regular
your interval in context.

mammograms in preventing deaths from breast cancer?

**22. Birthweight. **In 2003 the

*Journal of the American Medical*
b) If your conclusion is incorrect, what kind of error have

*Association *reported a study examining the possible im-
pact of air pollution caused by the 9/11 attack on New

**27. Pain. **Researchers comparing the effectiveness of two

York's World Trade Center on the weight of babies.
pain medications randomly selected a group of patients
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CHAPTER 22 Comparing Two Proportions
who had been complaining of a certain kind of joint pain.

**30. Retention rates. **In 2004 the testing company ACT,

They randomly divided these people into two groups,
Inc., reported on the percentage of first-year students at
then administered the pain killers. Of the 112 people in
4-year colleges who return for a second year. Their sam-
the group who received medication A, 84 said this pain
ple of 1139 students in private colleges showed a 74.9%
reliever was effective. Of the 108 people in the other
retention rate, while the rate was 71.9% for the sample of
group, 66 reported that pain reliever B was effective.

505 students at public colleges. Does this provide evi-
a) Write a 95% confidence interval for the percent of
dence that there's a difference in retention rates of first-
people who may get relief from this kind of joint pain
year students at public and private colleges?
by using medication A. Interpret your interval.

**31. Online activity checks. **Are more parents checking up

b) Write a 95% confidence interval for the percent of
on their teen's online activities? A Pew survey in 2004
people who may get relief by using medication B.

found that 33% of 868 randomly sampled teens said that
Interpret your interval.

their parents checked to see what Web sites they visited.

c) Do the intervals for A and B overlap? What do you
In 2006 the same question posed to 811 teens found 41%
think this means about the comparative effectiveness
reporting such checks. Do these results provide evidence
of these medications?
that more parents are checking?
d) Find a 95% confidence interval for the difference in
the proportions of people who may find these medica-

**32. Computer gaming. **Who plays online or electronic

tions effective. Interpret your interval.

games? A survey in 2006 found that 69% of 223 boys aged
e) Does this interval contain zero? What does that mean?
12–14 said they "played computer or console games like
f ) Why do the results in parts c and e seem contradictory?
Xbox or PlayStation . . or games online." Of 248 boys
If we want to compare the effectiveness of these two
aged 15–17, only 62% played these games. Is this evi-
pain relievers, which is the correct approach? Why?
dence of a real age-based difference?

**28. Gender gap. **Candidates for political office realize that

different levels of support among men and women maybe a crucial factor in determining the outcome of an elec-tion. One candidate finds that 52% of 473 men polled saythey will vote for him, but only 45% of the 522 women in

**JUST CHECKING **
the poll express support.

a) Write a 95% confidence interval for the percent of

**1. **We're 90% confident that if members are contacted

male voters who may vote for this candidate. Interpret
by e-mail, the donation rate will be between 4 and 18
your interval.

percentage points higher than if they received regular
b) Write and interpret a 95% confidence interval for the
percent of female voters who may vote for him.

c) Do the intervals for males and females overlap? What
Since a difference of 0 is not in the confidence interval,
do you think this means about the gender gap?
we'd reject the null hypothesis. There is evidence that
d) Find a 95% confidence interval for the difference in
more members will donate if contacted by e-mail.

the proportions of males and females who will vote

**3. **The proportion from the sample in 1995 has variabil-

for this candidate. Interpret your interval.

ity, too. If we do a one-proportion

*z*-test, we won't
e) Does this interval contain zero? What does that mean?
take that variability into account and our P-value will
f) Why do the results in parts c and e seem contradic-
be incorrect.

tory? If we want to see if there is a gender gap among

**4. **The difference in the proportions between the two

voters with respect to this candidate, which is the cor-
years has more variability than either individual pro-
rect approach? Why?
portion. The variance of the difference is the sum of

**29. Sensitive men.**
In August 2004,

*Time *magazine,
the two variances.

reporting on a survey of men's attitudes, noted that"Young men are more comfortable than older men talk-ing about their problems." The survey reported that 80of 129 surveyed 18- to 24-year-old men and 98 of 184 25- to 34-year-old men said they were comfortable.

What do you think? Is

*Time's *interpretation justified bythese numbers?

Source: http://statistics-ap-b-m-b.whs.westerly.k12.ri.us/modules/locker/files/get_group_file.phtml?gid=4939304&fid=28733447

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