# Statistics-Mathematics homework task

Statistics-Mathematics homework task

I want this to be done tomorrow around this time, Houston, Tx time. Please, make sure you can solve.

WORK 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 only 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 number of tr ials 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 number of bad results 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?