Question

Warning: This question involves a power calculation for a test with a two-sided alternative, and it is meaningful only if we do not wish to reach a directional conclusion in the hypothesis test (we intend to either reject H0 or not reject H0). Interpreting power can be tricky when the alternative hypothesis is two-sided and we want to reach a directional conclusion.)

3. Suppose we are about to sample 50 observations from a normally distributed population where it is known that σ = 12, but µ is unknown. We intend to test H0: µ = 15 against Ha: µ 6= 15 at α = 0.05.

(a) What values of the sample mean would lead to a rejection of the null hypothesis?

(b) What is the power of the test if µ = 16?

(c) What is the power of the test if µ = 17?

Answer 1

: µ = 15 vs : µ ≠ 15

Given : σ = 12 and n = 50 , Therefore = 12 / √ 50 = 1.6971

α = 0.05 , α/2 = 0.025

Therefore critical values = -1.96 and 1.96

a) = -1.96* +  µ = (-1.96*1.6971) + 15  

= 11.67

= 1.96* +  µ = (1.96*1.6971) + 15  

= 18.33

The sample mean values would lead to a rejection of the null hypothesis are 11.67 and 18.33

b) power of the test if µ = 16

Power = P ( Reject H0 , when it is false )

= 1 - P( 11.67 ≤    ≤ 18.33)

= 1 - [ P(   ≤ 18.33) - P(   ≤ 11.67) ]

=

= 1 - [ P( z ≤ 1.37) - P( z ≤ -2.55 ) ]

= 1 - [ 0.9147 - 0.0054 ]

= 0.0907

The power of the test if µ = 16 is 0.0907

(c) What is the power of the test if µ = 17

Power = P ( Reject H0 , when it is false )

= 1 - P( 11.67 ≤    ≤ 18.33)

= 1 - [ P(   ≤ 18.33) - P(   ≤ 11.67) ]

=

= 1 - [ P( z ≤ 0.78) - P( z ≤ -3.14) ]

= 1 - [ 0.7823 - 0.0008 ]

= 0.2185

The power of the test if µ = 17 is 0.2185