STAT 302 Midterm: STAT 302 UW Madison Solutions 05
Solutions to Homework 5
Statistics 302 Professor Larget
Textbook Exercises
4.79 Divorce Opinions and Gender In Data 4.4 on page 227, we introduce the results of a
May 2010 Gallup poll of 1029 US adults. When asked if they view divorce as “morally acceptable”,
71% of the men and 67% of the women in the sample responded yes. In the test for a difference in
proportions, a randomization distribution gives a p-value of 0.165. Does this indicate a significant
difference between men and women in how they view divorce?
Solution
If we use a 5% significance level, the p-value of 0.165 is not less than α= 0.05 so we would not
reject H0:pf=pm. This means the data do not show significant evidence of a difference in the
proportions of men and women that view divorce as “morally acceptable”.
4.82 Sleep or Caffeine for Memory? The consumption of caffeine to benefit alternateness
is a common activity practiced by 90% of adults in North America. Often caffeine is used in order
to replace the need for sleep. One recent study compares students’ ability to recall memorized
information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults
(between the ages of 18 and 39) were randomly divided into three groups and verbally given a list
of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half,
another group is kept awake and then given a caffeine pill an hour prior to testing, and the third
group is given a placebo. The response variable of interest is the number of words participants are
able to recall following the break. The summary statistics for the three groups are shown below
in the table. We are interested in testing whether there is evidence of difference in average recall
ability between any two of the treatments. Thus we have three possible tests between different
pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo.
Group Sample Size Mean Standard Deviation
Sleep 12 15.25 3.3
Caffeine 12 12.25 3.5
Placebo 11 13.70 3.0
(a) In the test comparing the sleep group to the caffeine group, the p-value is 0.003. What is
the conclusion of the test? In the sample, which group had better recall ability? According to
the rest results, do you think sleep is really better than caffeine for recall ability?
(b) In the test comparing the sleep group to the placebo group, the p-value is 0.06. What is the
conclusion of the test using a 5% significance level? Using a 10% significance level? How strong
is the evidence of a difference in mean recall ability between these two treatments?
(c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22. What is
the conclusion of the test? In the sample, which group had better recall ability? According to
the test results, would we be justified in concluding that caffeine impairs recall ability?
(d) According to this study, what should you do before an exam that asks you to recall
information?
Solution
(a) The p-value (0.003) is small so the decision is to reject H0and conclude that the mean recall
for sleep (¯xs= 15.25) is different from the mean recall for caffeine (¯xc= 12.25). Since the mean
for the sleep group is higher than the mean for the caffeine group, we have sufficient evidence to
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conclude that mean recall after sleep is in fact better than after caffeine. Yes, sleep is really better
for you than caffeine for enhancing recall ability.
(b) The p-value (0.06) is not less than 0.05 so we would not reject H0at a 5% level, but it is
less than 0.10 so we would reject H0at a 10% level. There is some moderate evidence of a differ-
ence in mean recall ability between sleep and a placebo, but not very strong evidence.
(c) The p-value (0.22) is larger than any common significance level, so do not reject H0. The
placebo group had a better mean recall in this sample (¯xp= 13.70 compared to ¯xc= 12.25), but
there is not enough evidence to conclude that the mean for the population would be different for a
placebo than the mean recall for caffeine.
(d) Get a good night’s sleep!
4.86 Radiation from Cell Phones and Brain Activity Does heavy cell phone use affect brain
activity? There is some concern about possible negative effects of radiofrequency signals delivered
to the brain. In a randomized matched-pairs study, 47 healthy participants had cell phones placed
on the left and right ears. Brain glucose metabolism (a measure of brain activity) was measured
for all participants under two conditions: with one cell phone turned on for 50 minutes (the “on”
condition) and with both cell phones off (the “off” condition). The amplitude of radio frequency
waves emitted by the cell phones during the “on” condition was also measured.
(a) Is this an experiment or an observational study? Explain what it means to say that this
was a “matched-pairs” study.
(b) How was randomization likely used in the study? Why did participants have cell phones on
their ears during the “off” condition?
(c) The investigators were interested in seeing whether average brain glucose metabolism was
different based on whether the cell phones were turned on or off. State the null and alternative
hypotheses for this test.
(d) The p-value for the test in part (c) is 0.004. State the conclusion of this test in context.
(e) The investigators were also interested in seeing if brain glucose metabolism was significantly
correlated with the amplitude of the radio frequency waves. What graph might we use to
visualize this relationship?
(f) State the null and alternative hypotheses for the test in part (e).
(g) The article states that the p-value for the test in part (e) satisfies p < 0.001. State the
conclusion of this test in context.
Solution
(a) This is an experiment since the explanatory factor (cell phone “on” or “off”) was controlled.
The design is matched pairs, since all 47 participants were tested under both conditions. For each
participant, we find the difference in brain activity between the two conditions.
(b) Randomization in this case means that the order of the conditions (“on” and “off”) was ran-
domized for all the participants. Cell phones were on the ears for both conditions to control for
any lurking variables and to make the treatments as similar as possible except for the variable of
interest (the radiofrequency waves).
(c) Using µon to represent average brain glucose metabolism when the cell phones are on and
µoff to represent average brain glucose metabolism when the cell phones are off, the hypotheses
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are:
H0:µon =µoff
Ha:µon 6=µoff
Notice that since this is a matched pairs study, we could also write the hypotheses in terms of the
average difference µDbetween the two conditions, with H0:µD= 0 vs Ha:µ6= 0.
(d) Since the p-value is quite small (less than a significance level of 0.01), we reject the null
hypothesis. There is significant evidence that brain activity is affected by cell phones.
(e) Both of these variables (brain glucose metabolism and amplitude of radiofrequency) are quan-
titative, so we use a scatterplot to graph the relationship.
(f) We are testing to see if the correlation ρbetween these two variables is significantly differ-
ent from zero, so the hypotheses are
H0:ρ= 0
Ha:ρ6= 0
where ρis the correlation between brain glucose metabolism and amplitude of radiofrequency.
(g) This p-value is very small so we reject H0. There is strong evidence that brain activity is
correlated with the amplitude of the radiofrequency waves emitted by the cell phone.
For 4.94 and 4.96, indicate whether it makes more sense to use a relatively large significance
level (such as α= 0.10) or a relatively small significance level (such as α= 0.01).
4.94 Using your statistics class as a sample to see if there is evidence of a difference between
male and female students in how many hours are spent watching television per week.
Solution
A Type I error (saying there’s a difference in TV habits by gender for the class, when actually there
isn’t) is not very serious, so a large significance level such as α= 0.10 will make it easier to see any
difference.
4.96 Testing to see if a well-known company is lying in its advertising. If there is evidence that
the company is lying, the Federal Trade Commission will file a lawsuit against them.
Solution
A Type I error (suing the company when they are not lying) is quite serious so it makes sense to
use a small significance level such as α= 0.01.
For 4.100 and 4.102, describe what it means in that context to make a Type I and Type II er-
ror. Personally, which do you feel is a worse error to make in the given situation?
4.100 The situation described in Exercise 4.94.
Solution
Type I error: Conclude there’s a difference in TV habits by gender for the class, when actually
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