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ECON 3100
PROBLEM SET 7
Textbook questions: Chapter 13: 3, 5, 11, 15, 19, 25
Statistical investigation:
This exercise calls on the data set “analgesics” (a JMP sample data set, available under the
“sample data sets” in JMP and on Canvas in the “Data sets” module). The data set contains the
following variables:
Gender: male or female
Drug: which of three drugs was administered, A, B, or C
Pain: the patient’s level of pain, with a higher value representing greater pain.
1. Create a table showing the average level of pain for each drug/gender combination. Put
“Gender” in the rows of the table and “Drug” in the Columns. What observations can you make
about differences in the effectiveness of the three drugs based on your table?
2. Conduct a one-way ANOVA analysis to test whether the average level of pain differs by
gender. Be sure to clearly state the null and alternative hypothesis, the test statistic, and the
conclusion. .
3. Conduct a one-way ANOVA analysis to test whether the average level of pain differs by the
drug taken. Be sure to clearly state the null and alternative hypothesis, the test statistics, and the
conclusion. .
4. Summarize your findings accurately and concisely in a brief paragraph.
Everyday Statistics
This question is based on the article “The Myth, the Math, the Sex,” which follows this question.
Suppose that you were conducting statistical research, using ANOVA to test for a difference in
the average number of sexual partners reported by men and the number reported by women.
1. What would be the response (dependent) variable in the study?
2. What would be the “treatment” or “factor” in the study?
3. What would be the null hypothesis?
4. Based on the information in the article and men’s and women’s survey responses, what would
you expect the conclusion to be?
5. Near the end of the article, Dr. Gale states, “the false conclusions people draw from these
surveys may have a sort of self-fulfilling prophecy.” Why must the conclusions people draw
from these surveys be false?
6. In your opinion, are people influenced by survey results- that is, do people shape their
behavior to match what survey results claim to be “normal” (or the opposite-do they try to stand
out from the norm)?
1
August 12, 2007
IDEAS & TRENDS
The Myth, the Math, the Sex
By GINA KOLATA
EVERYONE knows men are promiscuous by nature. It’s part of the genetic strategy that
evolved to help men spread their genes far and wide. The strategy is different for a
woman, who has to go through so much just to have a baby and then nurture it. She is
genetically programmed to want just one man who will stick with her and help raise
their children.
Surveys bear this out. In study after study and in country after country, men report
more, often many more, sexual partners than women.
One survey, recently reported by the federal government, concluded that men had a
median of seven female sex partners. Women had a median of four male sex partners.
Another study, by British researchers, stated that men had 12.7 heterosexual partners in
their lifetimes and women had 6.5.
But there is just one problem, mathematicians say. It is logically impossible for
heterosexual men to have more partners on average than heterosexual women. Those
survey results cannot be correct.
It is about time for mathematicians to set the record straight, said David Gale, an
emeritus professor of mathematics at the University of California, Berkeley.
“Surveys and studies to the contrary notwithstanding, the conclusion that men have
substantially more sex partners than women is not and cannot be true for purely logical
reasons,” Dr. Gale said.
He even provided a proof, writing in an e-mail message:
“By way of dramatization, we change the context slightly and will prove what will be
called the High School Prom Theorem. We suppose that on the day after the prom, each
girl is asked to give the number of boys she danced with. These numbers are then added
up giving a number G. The same information is then obtained from the boys, giving a
number B.
Theorem: G=B
2
Proof: Both G and B are equal to C, the number of couples who danced together at the
prom. Q.E.D.”
Sex survey researchers say they know that Dr. Gale is correct. Men and women in a
population must have roughly equal numbers of partners. So, when men report many
more than women, what is going on and what is to be believed?
“I have heard this question before,” said Cheryl D. Fryar, a health statistician at the
National Center for Health Statistics and a lead author of the new federal report, “Drug
Use and Sexual Behaviors Reported by Adults: United States, 1999-2002,” which found
that men had a median of seven partners and women four.
But when it comes to an explanation, she added, “I have no idea.”
“This is what is reported,” Ms. Fryar said. “The reason why they report it I do not know.”
Sevgi O. Aral, who is associate director for science in the division of sexually transmitted
disease prevention at the Centers for Disease Control and Prevention, said there are
several possible explanations and all are probably operating.
One is that men are going outside the population to find partners, to prostitutes, for
example, who are not part of the survey, or are having sex when they travel to other
countries.
Another, of course, is that men exaggerate the number of partners they have and women
underestimate.
Dr. Aral said she cannot determine what the true number of sex partners is for men and
women, but, she added, “I would say that men have more partners on average but the
difference is not as big as it seems in the numbers we are looking at.”
Dr. Gale is still troubled. He said invoking women who are outside the survey
population cannot begin to explain a difference of 75 percent in the number of partners,
as occurred in the study saying men had seven partners and women four. Something like
a prostitute effect, he said, “would be negligible.” The most likely explanation, by far, is
that the numbers cannot be trusted.
3
Ronald Graham, a professor of mathematics and computer science at the University of
California, San Diego, agreed with Dr. Gale. After all, on average, men would have to
have three more partners than women, raising the question of where all those extra
partners might be.
“Some might be imaginary,” Dr. Graham said. “Maybe two are in the man’s mind and
one really exists.”
Dr. Gale added that he is not just being querulous when he raises the question of logical
impossibility. The problem, he said, is that when such data are published, with no
asterisk next to them saying they can’t be true, they just “reinforce the stereotypes of
promiscuous males and chaste females.”
In fact, he added, the survey data themselves may be part of the problem. If asked, a
man, believing that he should have a lot of partners, may feel compelled to exaggerate,
and a woman, believing that she should have few partners, may minimize her past.
“In this way,” Dr. Gale said, “the false conclusions people draw from these surveys may
have a sort of self-fulfilling prophecy.”
4
Textbook questions: Chapter 13: 3, 5, 11, 15, 19, 25
3. Use the F table to determine the value beyond which you would find
a. 1% of the values, if numerator degrees of freedom = 3 and denominator degrees of
freedom = 13.
b. 5% of the values, if numerator degrees of freedom = 2 and denominator degrees of
freedom = 20.
c. 1% of the values, if numerator degrees of freedom = 3 and denominator degrees
of freedom = 17.
5. Use a statistical calculator or Excel's F.DIST.RT function to determine the proportion of
values in an F distribution that are greater than
1. 3.285, if numerator degrees of freedom = 4 and denominator degrees of freedom = 18.
2. 5.701, if numerator degrees of freedom = 1 and denominator degrees of
freedom = 12.
3. 7.238, if numerator degrees of freedom = 5 and denominator degrees of freedom =
28.
11. As sales reps for Nguyen Products, Byron and Anita have generally produced the
same average weekly sales volume, but the “consistency” (as measured by the variance of
weekly sales) of the two sales reps appears to be different. You take independent random
samples of six weeks of sales for each of the two reps. Results (in $1000s) are shown
below.
1.
Test the null hypothesis that the two populations represented here have equal
variances. Assume that both population distributions are normal. Use a significance level
of 10%.
2. Can you make the case from this sample data that Byron's sales are less
consistent (have a larger variance) than Anita's? That is, can we reject a
null hypothesis that the variance of Byron's sales is no greater than the
variance of Anita's sales? Use a significance level of 5%.
15. You have conducted recent tests measuring peak pollution levels in various parts of
the city. The average peak pollution readings for a sample of 21 days at each of three city
locations are given in the table below, along with sample standard deviations.
Test the hypothesis that the average peak pollution levels for the three populations
represented here are the same. Use a significance level of 5%.
19. Goveia Inc. is evaluating three possible bonus incentive programs for its sales staff.
During a trial period lasting four months, five sales staff members were randomly
assigned to each bonus program. Individual sales figures (in $millions) are shown below:
1.
Test the hypothesis that average sales for the three populations represented here
are the same. Use a significance level of 5%. (Use Expression 13.2 to compute the
within-groups sum of squares (SSW) for your analysis.)
2. Show the proper ANOVA table summarizing your work in part a.
25. Elam Industries is interested in assessing customer response to four possible national
promotional campaigns. Twenty retail stores were randomly selected for a preliminary
trial. The twenty stores were randomly divided into four groups of five stores and each
group was randomly assigned a different promotion. At the end of the trial period,
average response rates were calculated for each of the four groups. Complete the
ANOVA table below and use the information in the table to test a null hypothesis that the
average response rate would be the same for the four populations represented in the trial.
Use a significance level of .05. Report your conclusion and explain your reasoning.
Textbook questions: Chapter 13: 3, 5, 11, 15, 19, 25
3. Use the F table to determine the value beyond which you would find
a. 1% of the values, if numerator degrees of freedom = 3 and denominator degrees of
freedom = 13.
b. 5% of the values, if numerator degrees of freedom = 2 and denominator degrees of
freedom = 20.
c. 1% of the values, if numerator degrees of freedom = 3 and denominator degrees
of freedom = 17.
5. Use a statistical calculator or Excel's F.DIST.RT function to determine the proportion of
values in an F distribution that are greater than
1. 3.285, if numerator degrees of freedom = 4 and denominator degrees of freedom = 18.
2. 5.701, if numerator degrees of freedom = 1 and denominator degrees of
freedom = 12.
3. 7.238, if numerator degrees of freedom = 5 and denominator degrees of freedom =
28.
11. As sales reps for Nguyen Products, Byron and Anita have generally produced the
same average weekly sales volume, but the “consistency” (as measured by the variance of
weekly sales) of the two sales reps appears to be different. You take independent random
samples of six weeks of sales for each of the two reps. Results (in $1000s) are shown
below.
1.
Test the null hypothesis that the two populations represented here have equal
variances. Assume that both population distributions are normal. Use a significance level
of 10%.
2. Can you make the case from this sample data that Byron's sales are less
consistent (have a larger variance) than Anita's? That is, can we reject a
null hypothesis that the variance of Byron's sales is no greater than the
variance of Anita's sales? Use a significance level of 5%.
15. You have conducted recent tests measuring peak pollution levels in various parts of
the city. The average peak pollution readings for a sample of 21 days at each of three city
locations are given in the table below, along with sample standard deviations.
Test the hypothesis that the average peak pollution levels for the three populations
represented here are the same. Use a significance level of 5%.
19. Goveia Inc. is evaluating three possible bonus incentive programs for its sales staff.
During a trial period lasting four months, five sales staff members were randomly
assigned to each bonus program. Individual sales figures (in $millions) are shown below:
1.
Test the hypothesis that average sales for the three populations represented here
are the same. Use a significance level of 5%. (Use Expression 13.2 to compute the
within-groups sum of squares (SSW) for your analysis.)
2. Show the proper ANOVA table summarizing your work in part a.
25. Elam Industries is interested in assessing customer response to four possible national
promotional campaigns. Twenty retail stores were randomly selected for a preliminary
trial. The twenty stores were randomly divided into four groups of five stores and each
group was randomly assigned a different promotion. At the end of the trial period,
average response rates were calculated for each of the four groups. Complete the
ANOVA table below and use the information in the table to test a null hypothesis that the
average response rate would be the same for the four populations represented in the trial.
Use a significance level of .05. Report your conclusion and explain your reasoning.
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Notes Example of factorial experime
gender
drug
pain
Fit Model
1 male
C
12
2 female
A
8
3 male
C
14
4 female
A
10
5 male
C
12
6 female
А
9
7 female
A
7
8 female
A
2
Columns (370)
9 female
A
7
gender
10 female
A
6
drug
11 male
B
13
pain
12 female
C
12
13 female
A
7
14 male
C
14
15 male
A
8
16 male
B
17
17 female
с
9
18 male
C
Rows
11
33
19 female
B
8
Selected
1
20 male
А
7
Excluded
0
21 female
A
7
Hidden
0
22 male
B
10
Labelled
0
23 female
A
7
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A
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14 male
14
15 male
C
A
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8
17
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17 female
18 male
19 female
20 male
21 female
22 male
C
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gender
drug
pain
8
7
7
A
10
23 female
B
A
A
7
7
A
7
C
2
B
B
8
4
24 female
25 female
26 female
27 male
28 female
29 male
30 male
31 male
32 female
33 male
B
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