   # Consider the following hypothesis test. H 0 : μ = 20 H a : μ ≠ 20 A sample of 200 items will be taken and the population standard deviation is σ = 10.Use α = .05. Compute the probability of making a Type II error if the population mean is: a. μ = 18.0 b. μ = 22.5 c. μ = 21.0 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

#### Solutions

Chapter
Section 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 9.6, Problem 47E
Textbook Problem
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## Consider the following hypothesis test. H0: μ = 20 Ha: μ ≠ 20A sample of 200 items will be taken and the population standard deviation is σ = 10.Use α = .05. Compute the probability of making a Type II error if the population mean is: a. μ = 18.0 b. μ = 22.5 c. μ = 21.0

a.

To determine

Find the probability of making a Type II error if μ=18.

### Explanation of Solution

Calculation:

The given information is that n=200, μ=18, σ=10, and α=0.05.

The test hypotheses are given below:

Null hypothesis:

H0:μ=20

Alternative hypothesis:

Ha:μ20

The rejection rule for two tailed test using the critical value is,

If zzα or zzα, reject the null hypothesis

Critical value:

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability> OK.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tails for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical values are ±1.96.

Rejection rule:

If z1.96 or z1.96, reject the null hypothesis.

The value x¯1 is calculated as follows:

x¯1=μz(σn)=201.96(10200)=18

b.

To determine

Find the probability of making a Type II error if μ=22.5.

c.

To determine

Find the probability of making a Type II error if μ=21.

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