Download Homework 3, Statistics 512, Spring 2005

Homework 9, Statistics 512, Spring 2005
This homework is due Thursday, March 31st at the beginning of class.
1. Hogg, McKean and Craig, 6.5.3.
Note: For this “two sample” problem, the test statistic 2 log  converges in distribution
to a chi-squared distribution under the null hypothesis when both sample sizes m and
n converge to infinity (where the degrees of freedom equals the number of extra free
parameters in the alternative hypothesis compared to the null hypothesis).
2. In a classic genetics study, Geissler (1889) studied hospital records in Saxony and
compiled data on the gender ratio. The table below shows the number of male children in
6115 families with 12 children. Let X i denote the number of male children the i th
family has. Suppose that the probabilities of a family of 12 children having 0, 1, 2, ..., 12
male children are the same for each family and that the number of male children had by
each family is independent. Then we can regard X1 , , X 6115 as 6115 iid multinomial
random variables with probabilities p0  P( X i  0), , p12  P( X i  12) .
(a) Consider the model that the number of male children born to a family with 12 children
is a binomial random variable with p  0.5 and 12 trials. What multinomial probabilities
p0 , , p12 are implied by this model?
(b) Test the goodness of fit of the model in (a) using a likelihood ratio test, i.e., test the
null hypothesis that p0 , , p12 equal the values you found in (a) versus the general
multinomial alternative hypothesis that p0 , , p12 are not equal to the values from (a).
(c) Consider the model that the number of male children born to a family with 12 children
is a binomial random variable with unknown p and 12 trials. Test the goodness of fit of
this model using a likelihood ratio test statistic.
Number of Male
Frequency
Number of Male
Frequency
Children
Children
0
7
7
1033
1
45
8
670
2
181
9
286
3
478
10
104
4
829
11
24
5
1112
12
3
6
1343
3. Hogg, McKean and Craig, 7.2.2. Hint: See Theorem 3.2.1.
4. Suppose X 1 , X 2 , X 3 are iid Bernoulli random variables with success probability p . In
class we showed that Y  X1  X 2  X 3 is a sufficient statistic for p . Show that
Y '  X1  2 X 2  3 X 3 is not a sufficient statistic for p .