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Statistics 224/324, Spring 2007
Midterm Exam 2: Tuesday, March 27th, 2007
* Remember to write your name on the exam and the blue book.
* With the exception of the True/False and Multiple Choice, write all answers and work in the
blue books.
* Show all work to get full credit – partial credit will be awarded.
* When you are finished, tuck your exam into your blue book.
* Use your knowledge and skills to the best of your ability!
NAME_______________________
Your Points
True or False
Possible Points
6
Multiple Choice
8
Chuck Norris vs. Jack Bauer
20
Guitar Hero
10
More Than Meets the Eye
10
Air Pirates Hijacking an Airplane
Full of Green Bay Packers
15
TOTAL
69
Name___________________
1. True or False: Circle the Correct Answer (1 point each)
a. If X~Bin(n, p), Y ~ Bern(p), and X┴Y, then T = X+Y is distributed as Bin(n+1, p).
TRUE
FALSE
b. As n gets big and p gets small, in such a way that np goes to λ, the Binomial converges in
distribution to a normal random variable. TRUE
FALSE
c. For a hypothesis test using a continuous test statistic, and a fixed sample size, as the αlevel of the test decreases, the power of the test must decrease as well.
TRUE
FALSE
d. The p-value is defined to be the probability that the null hypothesis is true.
TRUE
FALSE
e. Between two unbiased estimators of the same population characteristic, generally the one
with the smaller variance is preferred. TRUE
FALSE
f. A Quincunx with only three horizontal rows of pegs would show an approximate normal
distribution at the bottom, as long as more than about 40 pellets were run through it.
TRUE
FALSE
g. To show that an estimator θ is the MVUE for some population parameter θ in a specific
sampling situation, you must show that θ is unbiased, and that θ has a smaller variance
than every other possible unbiased estimator that may exist for θ. TRUE
FALSE
2. Multiple Choice: Circle the Correct Answer (2 points each)
a. If X~Bin(200, 0.35), compute P(X ≤ 80) using the normal approximation to the binomial.
A. 0.0694
B. 0.4306
C. 0.9306
D. 0.9391
E. Normal approximation to the binomial is not appropriate here.
b. For a test of H0: θ = 10 vs. HA: θ ≠ 10, suppose our test statistic is U, and U ~ Unif(0, θ).
If the rejection region is defined to be U ≤ 2 or U ≥ 8, then what is the probability of a
Type I error (i.e., the α-level) for this test?
A. 0.0056
B. 0.2000
C. 0.2984
D. 0.4000
E. Cannot be calculated from given information
c. If the data is exactly normal, n = 10, xbar = 8, s2 = 5, and I compute a confidence interval
(5.702, 10.298), what is the confidence level I used?
A. 99.88%
B. 99.00%
C. 95.00%
D. 85.30%
E. 82.00%
d. Given 100 iid random variables Y1, Y2,…,Y100, with E(Yi) = 3.8 and VAR(Yi) = 6.2,
define T = Y1 + Y2 +…+Y100. Use the CLT to compute P(T ≥ 375).
A. Approximately 0.0000
B. 0.0222
C. 0.5832
D. 0.9778
E. CLT should not be used here
3. Chuck Norris vs. Jack Bauer (20 total points)
Chuck Norris and Jack Bauer are watching Animal Planet one day, when, after a show on
how octopi are as smart (or smarter) than a 3-year old human, Chuck challenges Jack to a battle
of wits. The goal is to come up with an estimator of the population mean, 1/ λ, based on a
random sample of size three from an exponential population. Consider each pick to be a random
variable Xi ~ iid Exp(λ). Jack, in a shameless plug for his show, suggests using:
e1 = (24*X1 + X2 + X3)/26.
Chuck thinks the first observation is crap, and thus prefers the estimator:
e2 = (0.5*X1 + X2 + X3)/3.
a. Compute the expected values of these two estimators. Are either of the estimators
unbiased for 1/λ? Hint: If X ~ Exp(λ), then what is E(X)? (6 points)
b. Compute the variances of these two estimators. Hint: If X ~ Exp(λ), then what is
VAR(X)? (6 points)
c. Compute the MSE for each of these estimators. Who won the battle of wits (i.e., whose
estimator had the better MSE)? (8 points)
4. Guitar Hero (10 total points)
A certain guitar hero (who will remain nameless) has a very unique act. During the
middle of the show, the hero swoops down from the ceiling wearing a harness, and ‘saves’ a girl
from the ‘roiling, rowdy, dance floor’. The harness and cable are rated to be able to sustain a
maximum load of 150lb beyond the weight of the hero. To insure the safety of himself and the
patrons, the guitar hero has hired a statistician to estimate the mean weight of the female patrons
at each show. The statistician does this by randomly sampling the weights of n = 13 individuals
as they enter the venue (there is a scale hidden in the floor of the entrance). If any part of a 99%
CI for the mean weight is larger than 150lb, the hero skips the rescue. You can assume that the
weights are independent, and that they are normally distributed.
a. Suppose at a particular concert, the statistician estimates xbar = 130 and s2 = 400.
Compute a 99% CI for the mean weight of the female patrons. Would the hero make the
rescue (i.e., is anything 150 or bigger in the confidence interval)? (4 points)
b. After several concerts where the statistician has declared the rescue unsafe, the hero
begins to wonder, “Am I just popular among the overweight female demographic, or is
the statistician not taking a large enough sample?” The hero confronts the statistician,
and poses the question, “Assuming the variance of the weights was known to be 441,
how many samples would you have to take in order to be able to estimate the true mean
weight to within ± 10lb with probability 99%?” In other words, the hero wants to know
the sample size required to get a 99% CI that has total width (from lower bound to upper
bound) of no more than 20lbs. Answer this question for the hero. (6 points)
5. More Than Meets the Eye (10 total points)
Optimus Prime is flipping through an armor catalogue, and can’t decide between two
different styles of breastplate. He decides to buy 5 of each, and test them out. He does this by
cutting each in half, and using half of style A and half of style B each time he enters into combat
with the evil Decepticons. After the battles, he rates the armor on a scale of 1 to 100 (100 being
completely undamaged). Here are the results after 5 battles:
Style A
Style B
Battle 1
Battle 2
Battle 3
vs.
vs.
vs.
Devastator Starscream Astrotrain
10
80
65
16
88
70
Battle 4
vs.
Soundwave
45
43
Battle 5
vs.
Shockwave
22
28
a. Assuming that the differences in the damage ratings are normally distributed, and that
the battles are independent, perform a paired t-test to determine if there is a difference
between the two styles using α = 0.05. State the null and alternative hypotheses,
compute the appropriate test statistic, and compute an approximate p-value. (10 points)
6. Air Pirates Hijacking an Airplane Full of Green Bay Packers (15 total points)
On route to a game in Chicago, a charter flight full of Packers is hijacked by air pirates disguised
as flight attendants. They put forth an ultimatum: If the average Packer can bench press 300lb,
they’ll let the plane land safely, and turn themselves in. Otherwise, they’ll plant a bomb, and
then parachute to safety. Since it’s a short flight, they randomly select 5 players with
replacement (hopefully not the kickers), and test them for their bench press. The pirates assume
that the bench-press values are normally distributed, and the variance of the bench presses is
known to be 5600. Letting μ be the true mean bench press weight, the pirates will test H0: μ =
300 vs. HA: μ ≠ 300, using the estimator Xbar, and decide they will blow up the plane if Xbar ≤
250 or Xbar ≥ 325. Notice that the rejection region is not symmetric around the null value!
(Those wily pirates.)
a. Compute the power of the test if the actual μ is 275. (6 points)
b. Determine whether the power would go up or down if the actual μ was instead 260.
Justify your answer. Hint: No additional computation is necessary. (3 points)
c. One of the Packers asks the pirates to take a larger sample size, since he feels that it will
improve the chances of his teammates proving themselves. Is this in the best interest of
the Packers? Answer this in the two possible cases: H0 true or H0 false. To restate:
i.
For a fixed rejection region, does the probability of rejecting the null hypothesis
increase or decrease when the sample size is increased, if the null is true? Justify
your answer. (3 points)
ii.
For a fixed rejection region, does the power (for any fixed alternative) increase or
decrease when the sample size is increased, if the null is false? Justify your
answer. (3 points)