   # 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 5. Use α = .05. If the actual population mean is 9, the probability of a Type II error is .2912. Suppose the researcher wants to reduce the probability of a Type II error to .10 when the actual population mean is 9. What sample size is recommended? ### Essentials Of Statistics For Busin...

9th Edition
David R. Anderson + 4 others
Publisher: South-Western College Pub
ISBN: 9780357045435

#### Solutions

Chapter
Section ### Essentials Of Statistics For Busin...

9th Edition
David R. Anderson + 4 others
Publisher: South-Western College Pub
ISBN: 9780357045435
Chapter 9.8, Problem 54E
Textbook Problem
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## Consider the following hypothesis test. H 0 : μ ≥ 10 H a : μ < 10 The sample size is 120 and the population standard deviation is 5. Use α = .05. If the actual population mean is 9, the probability of a Type II error is .2912. Suppose the researcher wants to reduce the probability of a Type II error to .10 when the actual population mean is 9. What sample size is recommended?

To determine

Find the sample size.

### Explanation of Solution

Calculation:

It is given that β=0.10, Power=0.90(=10.10), σ=5, μa=9 and μ0=10.

The null and alternative hypotheses are given below:

Null hypothesis:

H0:μ10

Alternative hypothesis:

Ha:μ<10

Sample size:

The formula for finding the sample size is as follows:

n=(zα+zβ)2σ2(μ0μa)2

Here, zα represents the z-value providing an area of α in the upper tail of a standard normal distribution, zβ represents the z-value providing an area of β in the upper tail of a standard normal distribution, σ represents the population standard deviation, μ0 represents the value of the population mean in the null hypothesis, and μa represents the value of the population mean used for the Type II error.

For zα:

It is given that α=0.05.

The cumulative area to the left is as follows:

Area to the left=1Area to the right=10.05=0.95

Critical value:

Use “Table 1: Cumulative probabilities for the standard normal distribution” to find zα

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