PA 818 Fall 2015
Professor Geoffrey Wallace
TA Wilson Law
Homework #8
(Due before discussions on Thurs November 12, 2015)
1. Picture Drawing. Given the following assumptions:
~
,
Draw pictures for each hypothesis tests below. Label your (1) distributions (and where they center),
(2) horizontal axes, (3) critical value, (4) rejection region, (5) probability of Type I, II error, power,
and as necessary and requested for each part.
a) If :
, significant level at , the true
, and the hypothesis test has a power of
0.50, what do the two distributions look like? Where is the sample mean ̅ that will give you a
p‐value of 2 ?
b) If :
, probability of committing Type I error is , and true
and the probability of
Type II error is 0.75.
′
c) Given the same test in part b), draw a graph with
where you will have 75% of statistical
power. On the same graph, draw the distributions where the sample size is now 2 but the test
and true remains the same.
d)
:
, where true
and a power of 0.75.
e) (Optional challenge question) Illustrate a 2‐tail hypothesis test in classical expressionism.
Realism or impressionism is also acceptable.
2. In October 16, 2009, the Rasmussen Reports surveyed 750 of potential GOP primary voters and
found that twenty‐nine percent (29%) of Republican voters nationwide say former Arkansas
Governor Mike Huckabee is their pick to represent the GOP in the 2012 Presidential campaign. They
found 24% prefer former Massachusetts Governor Mitt Romney while 18% would cast their vote for
former Alaska Governor Sarah Palin. Former House Speaker Newt Gingrich got 14% of the vote
while Minnesota Governor Tim Pawlenty got 4%. Six percent (6%) of GOP voters preferred some
other candidate while 5% remain undecided.
a) Describe the distribution of the sample proportion of GOP voters who would vote for Mike
Hukabee. A complete answer describes the distribution, mean and variance in terms of
population parameters.
b) Using the results of the Rasmussen Reports poll, construct a 95% confidence interval around the
proportion of GOP voters who would have voted for Huckabee on the day of the poll.
2
A similar poll of 750 likely GOP primary voters was conducted on July 6th, 2009. In this poll, 25% said
that they would vote for Romney, 24% said that they would vote for Palin, and 22% said they would
vote for Huckabee.
c) Describe the distribution of the difference in the sample proportion of Americans who would
vote for Huckabee in the two polls
. A complete answer describes the
distributional form, mean, and variance of the difference in sample proportions in terms of
population parameters. (Hint: you can think of the two polls as independent random sampling
of the two populations. One poll samples the population of GOP primary voters on July 6th,
while the other samples the population of Americans in mid‐October).
d) We see that the support for Huckabee appears to grow stronger from July to October, but is
there strong evidence to suggest that this is the case? Justify your answer using statistics you
learned from class. =)
3. There is a nationally representative dataset that documented the body image of middle school and
high school students. We know that higher proportion of high school boys have a poorer body
image than that of the middle school boys. We are interested to find out what’s the statistical
power of the test.
Supposewe found the sample proportion of high school and middle school boys who have poor
body image are respectively:
̅
0.10, ̅
0.08, ∆ ̅
̅
̅
0.02
And their respective sample sizes are:
600,
400.
a) What hypothesis we should construct to determine if higher proportion of high school boys
have poor body image than middle school boys?
b) What’s the distribution of the difference in means in population parameters? And what is the
distribution under the null hypothesis?
c) What’s the critical value for rejection? What’s the corresponding z‐value?
d) Suppose the true difference between the means are 0.04 (why is this important?), what is the
power of the test? What means should we assume?
e) We find that the power of the test is less than what we want. At the existing sample size, we
can’t increase the power without increasing Type I error at the same time. So we decide to
repeat the survey with larger sample size. Before we do that, we want to estimate the size we
need to see if we can afford it. So given the sample statistics above, at 95% significant level,
using the same sample size for both high school and middle school, what sample size do we
need to increase the power to 90%?
