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Answer :
Answer: sample required n = 18
Step-by-step explanation:
Given that the value under under null hypothesis is 40 while the value under the alternative is less than 40, specifically 35.9
∴ Hâ‚€ : u = 40
Hâ‚ : u = 35.9
therefore β = ( 35.9 - 40 ) = -4.1
The level of significance ∝ = 0.05
Probability of committing type 11 error P = 0.1
standard deviation α = 5.8
Therefore our z-vales (z table)
Zâ‚€.â‚… = 1.645
Zâ‚€.â‚ = 1.282
NOW let n be sample size
n = {( Zâ‚€.â‚… + Zâ‚€.â‚ )² × α²} / β²
n = {( 1.645 + 1.282 )² × 5.8²} / (- 4.1)²
n = 17.14485
Since we are talking about sample size; it has to be a whole number
therefore
sample required n = 18
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Rewritten by : Jeany
Final answer:
To determine the sample size required to have a probability of committing a type II error of 0.1, you use the power of the test. The formula is Sample Size = (Zα + Zβ)² * σ² / (μ0 - μ1)². Plugging in the values, a sample size of at least 16 is required.
Explanation:
To determine the sample size required to have a probability of committing a type II error of 0.1, we need to use the power of the test. The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.
In this case, the null hypothesis is that the true mean is 40 months, while the alternative hypothesis is that the true mean is less than 40 months. The true mean provided is 35.9 months. To calculate the sample size, we need to use the formula:
Sample Size = (Zα + Zβ)² * σ² / (μ0 - μ1)²
Where Zα is the critical z-value for the given level of significance, Zβ is the critical z-value for the desired power of the test, σ is the population standard deviation, and μ0 and μ1 are the mean under the null and alternative hypotheses, respectively.
Since the level of significance is 0.05, the critical z-value for a one-tailed test is -1.645. The critical z-value for a power of 0.9 is approximately 1.28.
Plugging in the values, we get:
Sample Size = (1.645 + 1.28)² * 5.8² / (40 - 35.9)²
Sample Size ≈ 15.39 ≈ 16
Therefore, a sample size of at least 16 is required to have a probability of committing a type II error of 0.1 when the true mean is 35.9 months.
