1.     A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency   test. The professor wants the class to be able to score above 70 on the test. The six students get scores of 92, 62, 95, 69, 82, and 75.

a)    Suppose that the math scores of her students have a Normal(µ, 102) distribution. Use the following uniform prior: define 41 hypotheses where u is 60, 61,  …, 100, and is    equally likely to be any of these values.

i)     Calculate the posterior distribution of u, and plot it.

ii)   What is the mean of the posterior?

iii)  Give a 95% credible interval for u.

iv)  What is the probability that u is greater than 70?

b)    Now suppose that the national average is 75 with standard deviation 15. Repeat part

a) using a Normal(75, 152) prior for u.

2.     Now suppose that the math scores of her students have a Normal(70, 2).  Use the same sample of data from Question 1.

a)    Use the following uniform prior: define 100 hypotheses where  is 1, 2,  …, 100, and is equally likely to be any of these values.

i)     What is the posterior distribution of ? Plot this posterior.

ii)   What is the mean of this posterior distribution?

iii)  Give a 95% credible interval for  .

iv)  What is the probability that  is greater than 10?

b)    Now suppose we put a Gamma(1, 100) prior on T = 1/2 . Repeat part a) using this prior.