Question

13. (Bonus, 10 points) 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 apply a z-test. The hypotheses are Ho: u=12.2 seconds and Ha: u<12.2 seconds. Assume that o=0.35 seconds, n=30. If we set the significance level a to be 0.10 (that is, fix the probability of making Type 1 error to be 0.1), please find the probability of making a Type ll error if in fact u=12.1 seconds. Hint: First find the decision bound under Ho: u=12.2, and then determine the area to the right of the bound using the true u = 12.1. Definition of Type I error: Rejecting the null hypothesis when it is in fact true. Definition of Type II error: Not rejecting the null hypothesis when it is in fact false.

Answer 1

The values of sample mean X̅ for which null hypothesis is rejected
Z = ( X̅ - µ ) / ( σ / √(n))
Critical value Z(α/2) = Z( 0.1 /2 ) = ± 1.282
1.282 = ( X̅ - 12.2 ) / ( 0.35 / √( 30 ))
Rejection region X̅ <= 12.12

Z = ( X - µ ) / ( σ / √(n))
Z = ( 12.12 - 12.1 ) / ( 0.35 / √ ( 30 ) )
Z = 0.313

P ( X̅ > 12.12 ) = 1 - P ( Z < 0.31 )
P ( X̅ > 12.12 ) = 1 - 0.6229

P ( X̅ > 12.12 | µ = 12.1 ) = 0.3771

P ( Type II error ) ß = 0.3771