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?

Q: Expert Answer

To determine Find the sample size. Explanation Calculation: It is given that β = 0.10 , Power = 0.90 ( = 1 − 0.10 ) , σ = 5 , μ a = 9 and μ 0 = 10 . The null and alternative hypotheses are given below: Null hypothesis: H 0 : μ ≥ 10 Alternative hypothesis: H a : μ < 10 Sample size: The formula for finding the sample size is as follows: n = ( z α + z β ) 2 σ 2 ( μ 0 − μ a ) 2 Here, z α represents the z -value providing an area of α in the upper tail of a standard normal distribution, z β represents the z -value providing an area of β in the upper tail of a standard normal distribution, σ represents the population standard deviation, μ 0 represents the value of the population mean in the null hypothesis, and μ a represents the value of the population mean used for the Type II error. For z α : It is given that α = 0.05 . The cumulative area to the left is as follows: Area to the left = 1 − Area to the right = 1 − 0.05 = 0.95 Critical value: Use “Table 1: Cumulative probabilities for the standard normal distribution” to find z α

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David R. Anderson + 4 others

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David R. Anderson + 4 others

Publisher: South-Western College Pub

ISBN: 9780357045435

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To determine

Find the sample size.

Explanation of Solution

Calculation:

It is given that β=0.10, Power=0.90(=1−0.10), σ=5, μa=9 and μ0=10.

The null and alternative hypotheses are given below:

Null hypothesis:

H0:μ≥10

Alternative hypothesis:

Ha:μ<10

Sample size:

The formula for finding the sample size is as follows:

n=(zα+zβ)2σ2(μ0−μa)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=1−Area to the right=1−0.05=0.95

Critical value:

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

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