Chapter 8
Problem Solving
E7
A sample of 549 randomly selected teenagers ages 1317 were asked whether it is appropriate for parents to
install a computer program limiting what teens can
access on the Internet. Fifty-two percent responded
that, yes, this was an appropriate parental measure.
Check the conditions and then construct a 90%
confidence interval for the proportion of teenagers
nationwide who would think this is an appropriate
measure.
E25
Answer-D
E27
Suppose 22 students out of a random sample of 40
students carry a backpack to school. Follow steps ac to test the claim that 60% of the students in the
school carry a back pack.
a. State the null hypothesis in words and
symbols
b. Calculate the value of the test statistic and
p-value for the test. Use the P-value in a
sentence that explains what it represents.
c. What is your conclusion? Explain it in the
context of this problem.
E33
Suppose that when flipping a penny, the
probability of getting heads is 0.5. Suppose 100
people perform a test of significance with this
penny. Each person plans to reject the null
hypothesis that p=0.5 if their P-value is less than
0.05. How many of these people do you expect
to make a Type I Error?
E35
Jeffery and Taline each want to test whether the
proportion of adults in their state who have
graduated from college is 0.6, as claimed in the
newspaper. Jeffery takes a random sample of 200
adults and uses α=0.01. Taline takes a random
sample of 200 adults and uses α=0.05. Suppose the
newspaper’s percentage is actually right.
a. Is it possible for Jeffery or Taline to make a
Type I Error? If so, who is more likely to do so?
b. Is it possible for Jeffery or Taline to make a
Type II Error? If so, who is more likely to do so.
E39
In P24 on page 510, you computed a test statistic for this study: Forty-five
dogs and their owners, chosen at random, were photographed separately.
A judge was shown a picture of each owner and pictures of two dogs and
asked to pick the dog that went with the owner. The judge was right 23
times.
a. At the α=0.05 do you reject the hypothesis that, in the long run,
the judge will select the correct dogs half the time?
b. Suppose you compute the 95% confidence interval for the
proportion of times the judge will select the correct dogs. Will the
value of p0 from the null hypothesis be in the confidence interval?
Explain.
c. Compute the confidence interval to check your answer in part b.
d. Explain how the margin of error for the confidence interval is
related to the power of the test.
e. How would you suggest that the investigator get more power for
the test?
E41
At the beginning of this section, you read that 2% of the barn
swallows have white feathers in places where plumage is
normally blue or red. However, about 16% of barn swallows
captured around Chernobyl after 1991 had such genetic
mutations. The number captured around Chernobyl was
relatively large – 266 barn swallows.
a. If you want to determine whether the increase in the
proportion of mutations is statistically significant, should
this be a one-sided or two sided test?
b. What conditions are not satisfied?
c. The display shows the sampling distribution of the
sample proportion, p-hat, for n=266 and p=0.02. Use it to
estimate the P-value for the test you selected.
d. What should the researchers conclude?
E47
A Statistics instructor kept track of students’ grades on her final exam and
whether they did the exam in pencil or in pen. Of the 75 students who used
pencil, 63 got grades of 70% or higher. If the 19 students who used pen, 10
got grades of 70% or higher. Her 95% confidence interval for the difference is
(0.07, 0.55). Assume the these students can be considered random samples
from the students who could take this instructor’s final. Which of these
choices are the best two interpretations of this confidence interval?
A. the teacher must observe more students, because the confidence
interval is too wide to be any possible value.
B. The teacher has evidence that a larger proportion of students who use a
pencil score 70% or higher on her final than students who use a pen.
C. If the teacher could observe all students who could take her final, she is
95% confident that the difference in the proportion who use a pencil and
get a grade 70% or higher and the proportion who use a pen and get a
grade of 70% or higher is between 0.07 and 0.55.
D. If the teacher could observe all students who could take her final, she is
95% confident that between 7% and 55% of them will get a 70% or
higher.
E49
Researchers at the University of California, San Diego, found
that when given a choice of two dogs, Judges correctly
matched 16 out of 25 purebred dogs with their owners.
When the dogs were mutts, only 7 dogs out of 20 were
correctly matched with their owners.
a. Check the conditions for constructing a confidence
interval for the difference of two proportions.
b. Find the 95% confidence interval for the difference of
proportions.
c. Interpret the resulting interval in context of the problem.
d. Is 0 in the confidence interval? What does your answer
imply?
e. What would recommend that the researchers do?
E51
A Harris poll asked this question of about 425 men and 425 women:
“When you get a sales or customer service phone call from someone
you don’t know, would you prefer to be addressed by your first name
or by your last name, or don’t you care one way or the other?”
Twenty-three percent of the man and 34% of the women said they
would prefer being addressed by their last name. Although this was
not quite the case, you can assume that the samples were random and
independent.
a. Check the conditions for constructing a confidence interval for the
difference of two proportions.
b. Find the 99% confidence interval for the difference of proportions.
c. Interpret the resulting interval in the context of the problem.
d. Is 0 in the confidence interval? What does your answer imply?
E53
In a recent national survey, 15, 200 high school
seniors in 128 schools completed questionnaires
about physical activity. Male students were
significantly more likely (59%) than female students
(48%) to have played on sports teams run by their
school during the 12 months preceding the survey.
Check the accuracy of the phrase “significantly
more likely” assuming that there were equal
numbers of male and female students in this survey
and that the samples are equivalent to independent
simple random samples.
E57
The formula for the confidence interval for the
difference of two proportions involves z. Why is
it ok to use z in this case?
E59
A Harris online poll asked 99 online respondents, “Would you
support the proposed bill that would eliminate the U.S.
penny?” There were 22 (or 22.2%) yes votes and 77 (of
77.8%) no votes. A student computed a 95% confidence
interval for the difference of the proportion of the population
answering yes and the proportion answering no as follows:
(.222)(.778) (.778)(.222)
(.222 - .778) ±1.96
+
99
99
= -.556 ± .116
Assuming the respondents can be considered a random
sample from a population, is the students’ method correct? If
so, write an interpretation of this interval. If not, do a more
appropriate analysis.
E61
Milk Chocolate M&M’s are 24% blue, and Almond M&M’s are
20% blue. Suppose you take a random sample of 100 of each.
You compute the proportion that are blue in each sample.
You then subtract the proportion of the blue in the sample of
Almond M&M’s from the proportion of blue in the sample of
Milk Chocolate M&M’s to get pˆ1 - pˆ 2. You repeat this process
until you have a distribution of millions of values of pˆ1 - pˆ 2.
a. What is the expected value of pˆ1 - pˆ 2 ?
b. What is the standard error of the distribution pˆ1 - pˆ 2 ?
a. Sketch the distribution of pˆ1 - pˆ 2 , with a scale on the
horizontal axis.
b. Compute the probability that the difference will be 0.05
or greater.
E63
A study of newly diagnosed lung cancer patients who had
enrolled in an Early Lung Cancer Action program regimen of
screening found that 234 of the 258 patients with tumors
measuring 15mm or less had no metastases (spread to other
body parts) and that 98 of the 118 patients with tumors
measuring 16-25mm had no metastases. The researchers
needed to determine whether the difference in these two
proportions is statistically significant.
a. The investigators chose to do a one-sided test of
significance. Why does this make sense?
b. Are the conditions for inference met?
c. The P-value reported in the article was 0.02. Is that correct?
d. Interpret this P-value in the context of the situation.
E65
A poll of 256 boys and 257 girls ages 12-17 asked, “Do you feel
like you are personally making a positive difference in you
community?” More girls (76%) than boys (63%) answered yes.
a. Based on using a one-sided test, is this a statistically
significant difference? That is if all teens were asked, are you
confident that a larger proportion of girls that boys would
answer yes? Assume that the samples were selected
randomly.
b. The report says, “Participants were selected through random
digit dialing.” Do you have any concerns about whether such
a procedure would give a random sample?
c. Find the 95% confidence interval for the proportion of all
teens who would answer yes. What additional assumptions
do you need to make to do this?
E69
What is the trend in Internet use? National polls of
about 1000 adults each estimated that 47% of
adults logged on to the Internet for an hour or
more daily in 2003 and that increased to 51% by
2005.
a. Is this a statistically significant increase?
b. Are you justified in saying that a majority of
adults in the United States logged on to the
Internet daily in 2005? Explain your reasoning.
E71
You feed Diet A to a random sample of cows and Diet B to another random
sample. Your null hypothesis is that the two diets produce the same
proportion of contented cows. Your two-sample test has a P-value of 0.36.
Select the one best interpretation of this P-value.
A.
B.
C.
D.
E.
If the null hypothesis that there is no difference in the proportion of contented cows
produced by the two diets is true, the probability is 0.36 that the proportion of contented
cows produced by each diet won’t be very different.
You can’t conclude that the diets make a difference because it’s fairly likely to get as big as
difference in the proportions of contented cows produced by the two diets as you did even
if there is no difference at all in the overall proportion of contented cows that each diet
produced.
The probability is 0.36 that the two diets differ in the proportion of contented cows that
they produced. Because this probability is so high, you can reject the null hypothesis.
If the null hypothesis is true, then the probability is 0.36 that the two diets differ in the
proportion of contented cows.
You can’t reject the null hypothesis because if, in fact, the null hypothesis is true, the
probability is 0.36 that the difference in the true proportion is the same as in your
experiment.