   # consider the following hypothesis test. H 0 : μ ≥ 10 H a : μ &lt; 10 The sample size is 120 and the population standard deviation is assumed known with σ = 5. Use α = .05. a. If the population mean is 9, what is the probability that the sample mean leads to the conclusion do not reject H 0 ? b. What type of error would be made if the actual population mean is 9 and we conclude that H 0 : μ ≥ 10 is true? c. What is the probability of making a Type II error if the actual population mean is 8? 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

#### Solutions

Chapter
Section 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 9.6, Problem 46E
Textbook Problem
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## consider the following hypothesis test. H0: μ ≥ 10 Ha: μ < 10 The sample size is 120 and the population standard deviation is assumed known with σ = 5. Use α = .05. a. If the population mean is 9, what is the probability that the sample mean leads to the conclusion do not reject H0? b. What type of error would be made if the actual population mean is 9 and we conclude that H0: μ ≥ 10 is true? c. What is the probability of making a Type II error if the actual population mean is 8?

a.

To determine

Find the probability that the sample mean leads to the conclusion do not reject H0.

### Explanation of Solution

Calculation:

The given information is that the sample size is n=120, the population standard deviation is σ=5, μ=9 and α=0.05.

The hypotheses are given below:

Null hypothesis:

H0:μ10

Alternative hypothesis:

Ha:μ<10

The rejection rule for lower tail test using the critical value is,

If 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 Left Tail 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 value is −1.645.

Rejection rule:

If z1.645, reject the null hypothesis

The value x¯ is,

x¯=μz(σn)=101

b.

To determine

Identify the type of error would be made if the actual population mean is 9 and conclude that H0:μ10 is true.

c.

To determine

Find the probability of making a Type II error if the actual population mean is 8.

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