For the standard normal distribution function , use the table that the National Institute of

Standards and Technology (NIST) provides at www.itl.gov/div898/handbook/eda/section3/eda3671.html

(sorry for the ridiculous link. A clickable line is in the desciption of the assignment. It's also linked

in the References section of the Wikipedia page on the standard normal distribution.)

Exercise 1. Suppose you ip a biased coin (with probability of heads p = 0.4) n = 100 times. What

is the probability of at most 90 heads? Express this in terms of a cumulative binomial distribution

function.

Exercise 2. Give an approximate value for F100,0.4(90) using the NIST table for the standard

normal cumulative distribution function .

Exercise 3. What is the p-value for the outcome of 90 heads in the experiment of ipping a coin

(independently) n = 100 times, when the null hypothesis is the coin has probability of heads

p = 0.4? How does this relate to a decision rule and Type I error probability?

Exercise 4. Joe Labcoat says that the pvalue is the probability that the null hypothesis is true.

In the exercise above, is this borne out? Why or why not?

Exercise 5. In a conventional test of hypotheses, is there an event E which does not depend on

the outcome of the experiment for which the p-value is equal to P(E)? What event is that?

Exercise 6. Suppose you test the hypothesis the coin is biased with probability p = 0.4 versus

the alternative the coin is fair using the Bayesian approach. Fill in the table below with the

(posterior) probability that the null hypothesis is true for the various prior probabilities given.

prior probability of null posterior probability of null

0.1

0.3

0.5

0.7

0.9