Question

Consider a general one-sided hypothesis test on a population mean µ with null hypothesis H0 : µ = 0, alternative hypothesis Ha : µ > 0, and Type I Error α = 0.02. Assume that using a sample of size n = 100 units, we observe some positive sample mean x > 0 with standard deviation s = 5. (a) Calculate the Type II Error and the power of the test assuming the following observed sample means: (i) x = 1.5 and (ii) x = 2.0. (b) How does the power of test behave as the observed sample mean x gets further away from the null hypothesis mean µ0?

Answer 1

Consider a general one-sided hypothesis test on a population mean µ with null hypothesis H0 : µ = 0, alternative hypothesis Ha : µ > 0, and Type I Error α = 0.02. Assume that using a sample of size n = 100 units, we observe some positive sample mean x > 0 with standard deviation s = 5.

1. Calculate the Type II Error and the power of the test assuming the following observed sample means: (i) x = 1.5

Power= 0.8197

Type II Error = 1- power = 1-0.8197 = 0.1803

2. and (ii) x = 2.0.

Power= 0.9715

Type II Error = 1- power = 1-0.9715 =0.0285

MINITAB result:

Power and Sample Size

1-Sample t Test

Testing mean = null (versus > null)

Calculating power for mean = null + difference

α = 0.02 Assumed standard deviation = 5

Results

Difference	Sample Size	Power
1.5	100	0.819687
2.0	100	0.971502

(b) How does the power of test behave as the observed sample mean x gets further away from the null hypothesis mean µ0?

Power of test increases as the observed sample mean x gets further away from the null hypothesis mean µ0.