Statistics

W O R K 1 Suppose you are testing a hypothesis H�0��
, and your

background assumptions X �
imply that the only alternative hypothesis to H�0 ��
is H�1��
. Assume that initially you are indifferent between H�0 ��
and H�1��
. Imagine that you will reject H�0 ��
if a certain test statistic falls in a rejection region. Let R �
be the proposition "The test statistic falls in the rejection region." Assume you have P�[�R�|��H�0��X�]�=�0.05� 
, as usual. However, assume that you also have P�[�R�|��H�1��X�]�=�0.04�
. What would be the correct posterior probability assignment P�[�H�0��|��R�X�]�
? Can you think of a real-world example in which this might happen?

2 . Consider the “optional stopping" problem with this setup:

Adam performs a series of independent experiments each with a Good (G) or Bad (B) outcome and obtains the data, D�=� 
 "The number of trials was 12 and the number of bad results was 3."

Assume the o n l y possible hypotheses are H�0��=� 
 “The success rate of obtaining good results is 50%.” and H�1��=� 
 "The success rate of obtaining good results is 75%."
Assume a prior probability assignment for H�0��
 of 10%. 


a) First, assume the background assumptions X�
 specify that the n u m b e r o f t r i a l s was fixed at 12 (Adam was going to do 12 trials no matter what). Compute the posterior probability P�[�H�0��|��D�X�]�
.

b) Now, assume instead that the background assumptions X�′��
 specify that the n u m b e r o f b a d r e s u l t s was fixed at 3 (Adam was going to keep going until he obtained 3 bad results). Compute the posterior probability P�[�H�0��|��D�X�′��]�.� 
 (Hint: You will need to use the negative binomial distribution for this calculation.)

c) Does the result surprise you?