Statistics: Determine probability of type II error?
After training intensively for six months, John hopes that his mean time to run 100 meters has decreased from last year’s mean time of 12.2 seconds. He performs a hypothesis test to determine whether his mean time has decreased. Preliminary data analyses indicate that it is reasonable to a apply a z-test. The hypotheses are –
null: µ = 12.2 seconds
alternate:µ < 12.2 seconds
Assume that sigma = 0.35 seconds, n = 30, and the significance level is 0.10. Find the probability of a Type II error if in fact µ = 12.1 seconds.
Would somebody be willing to help walk me through how to run this on a TI-83/84 calculator?
Thanks for any help you can provide.
Type II error is failing to reject the null hypothesis when you should have.
The cutoff for rejecting the null hypothesis is 12.2-1.28*0.35/sqrt 30 = 12.12.
So the null hypothesis is not rejected if the sample mean is greater than 12.12.
Assuming John’s mean time improved to 21.1 seconds, the probability of type II error is P(sample mean > 12.12) = P(z > (12.12-12.1) / (0.35/sqrt 30) )
=P(z > 0.31) = 0.3783.
I don’t know how the calculator works but this is how to solve the problem.