Question

Suppose that a basketball player (let’s call her Tamika) has had a .6 probability of successfully

making a foul shot (trial) during her career. Suppose further that Tamika wants to convince the

coach that she has improved to the extent that her probability of success is now .8 rather than .6.

The coach agrees to allow Tamika to take 50 shots in an effort to convince him that she has

improved to this extent. Let p represent Tamika’s actual probability of success. If p = .6, then

Tamika has not improved. If p = .8, then Tamika has improved to the extent she claimed.

Let Y represent the number of shots that she successfully makes. Consider decision rules which

say that the coach will be convinced that Tamika has improved if she successfully makes at least

c shots.

g) Determine the smallest value of c for which the probability of Type I error is .05 or smaller.

Also report the probability of Type I error with this value of c.

h) For the value of c that you determined in the previous question, determine the probability of

Type II error.

i) Describe the effect of a larger sample size on the error probabilities.

j) Re-answer questions g) and h), after changing the .05 criterion to a .10 criterion. Also

comment on what effect this change has on the probabilities of Type I error and Type II error.

Answer 1

We'll be using R for solving the problems here

g) Here we need to find the value of c such that P(X<c)<= 0.05, where X follows Binomial(50,0.8). Using R we're getting c=35. Also, the type 1 error here is coming to 0.05.

h) Here under H1 X follows Binomial(50,0.6), Thus we need to calculate P(X>35) which is 0.054

i) As we increase the sample size, there will be a decrease in the error probabilities.

j) (1) Here we need to find the value of c such that P(X<c)<= 0.1, where X follows Binomial(50,0.8). Using R we're getting c=36. Also, the type 1 error here is coming to 0.05.

(2)Here under H1 X follows Binomial(50,0.6), Thus we need to calculate P(X>36) which is 0.028