Math 152 - Midterm 2
You can use wolfram alpha to do any calculus (though I don’t think you’ll need
to), but otherwise are limited to your notes and the text book. Stop working
48 hours after starting. Due by the end of Wednesday.
1. Consider the following random sample coming from a continuous pdf that is
symmetric about : 1, 3, 4, 6, 10, 15. Consider a test for H0 : = 0 based on a
p-value coming from a binomial distribution, compute a 95 percent con dence
interval for .
2. Consider testing H0 : p = :5 vs Ha : p>:5 using a decision rule of the form
\reject H0 if X k", where X Bin(10;p).
a. It seems reasonable to only consider values of k > 5. For k = 6;:::;10,
calculate the value corresponding to each of these tests.
b. Notice that there does not exist a value k for which = :05. This is often
the case when our random variable is discrete. We can create a rule that will
have = :05 by using a randomized decision rule. For some values of X, we
know to reject. For others, we know to accept H0. For some values, we \ ip a
coin", the outcome of which tells us how to conclude. Give such a rule that has
= :05 (you need to tell me for what value you need to ip the coin and what
the probability of heads for the coin needs to be).
3. Consider a shifted exponential distribution with = 1 known.
fX(xj ) = e (x )1(x )
a. Derive the likelihood ratio test for H0 : = 2 vs Ha : 6= 2:
b. For n = 2, give the rejection region corresponding to an = :05.
c. Using the rejection region given in b, invert it to construct a con dence
interval for .
4. a. Consider trying to use a ‘goodness-of- t’ test for assessing the normality
of a random sample (this test is the common name for our 2 approximation
to the GLRT for multinomial data, recall free-throws/Hardy-Weinberg from the
homework). We decide to group the data into 3 bins to make it look multinomial:
12 observations fell less than 5, 64 fell between between 5 and 10, and 24 fell
above 10. The null hypothesis is simply that the data is normal. Argue that
this test is not feasible, and calculate the MLE for the parameters of the normal
(you can just use intuition if you’d like, I just want the right answer).
b. (Hint: The 2 distribution describes a continuous random variable.) For two
sided tests of the form. H0 : = 0 vs H1 : 6= 0, we have 2 log ! 2
under smoothness conditions. Argue that a two sided test of the form. H0 : 2
( 0 ; 0 + ) vs the two sided alternative cannot have this limiting behavior
( > 0).
5. Science typically dictates that empirical results be established at an = .05
level for our statistical test. Suppose that we believe that half of all comets in
the universe emit particles with a corresponding rate of = 2 (according to a
Poisson distribution) and the other half have rate of = 1. Researchers will
record the time until the 5th particle is emitted, and are interested in doing a
classi cation problem (guessing what type of comet they are observing based
on data). They decide to do this by testing H0 : = 1 vs H1 : = 2 and use
= :05. What are they implicitly saying about how many times worse a type
1 error is relative to a type II error?
6. Recall from homework the density f(x) = 12(1 + x) for x 2 [ 1;1] and
2 [ 1;1]. Consider the data set (n = 5): x = (:2;:4;:6;:8;:9). Compute the
p-value using the asymptotic result for the GLRT for the test H0 : = 0 vs
H1 : 6= 0. Compare this to the p-value coming from an (intuitive?) exact
test similar to the one in problem 1. (To gure out the MLE, consider the case
where n = 2, any subset of the data goes, and extrapolate.)