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To solve this problem, we need to find the required sample size \( n \) for a hypothesis test given specific Type I and Type II error probabilities. Here’s a breakdown of the information provided and the solution steps:

### Given Information
1. Null hypothesis: \( H_0: \mu = 40 \)
2. Alternative hypothesis: \( H_1: \mu > 40 \)
3. Standard deviation of the population (\( \sigma \)) = 4
4. Probability of Type I error (significance level, \( \alpha \)) = 0.03
5. Probability of Type II error (\( \beta \)) = 0.08 for \( \mu = 39 \)

### Step-by-Step Solution
1. **Determine the Z-scores** for Type I and Type II errors using standard normal distribution tables.
   - For \( \alpha = 0.03 \), the corresponding Z-score (right-tail) is approximately \( Z_{\alpha} = 1.88 \).
   - For \( \beta = 0.08 \), the corresponding Z-score (left-tail for power calculation) is approximately \( Z_{\beta} = 1.41 \).

2. **Calculate the Difference in Means**:  
   Since the alternative hypothesis considers \( \mu > 40 \), we calculate the sample size assuming the mean could be 39 (as specified in the question). The difference in the mean is:
   \[
   \Delta = 40 - 39 = 1
   \]

3. **Set up the sample size formula**:  
   The formula for sample size \( n \) when testing for a difference in means is:
   \[
   n = \left( \frac{Z_{\alpha} + Z_{\beta}}{\frac{\Delta}{\sigma}} \right)^2
   \]

   Substituting the values:
   \[
   n = \left( \frac{1.88 + 1.41}{\frac{1}{4}} \right)^2
   \]

4. **Calculate the sample size \( n \)**:
   - Calculate the combined Z-score: \( 1.88 + 1.41 = 3.29 \)
   - Divide by the standardized effect size: \( \frac{3.29}{\frac{1}{4}} = 3.29 \times 4 = 13.16 \)
   - Square the result to find \( n \):
     \[
     n = (13.16)^2 = 173.1856
     \]

   Rounding up (since sample size must be a whole number), we find:
   \[
   n \approx 174
   \]

### Final Answer
The required sample size \( n \) is approximately **174**.

Would you like a detailed explanation of each step or have any questions?

### Related Questions
1. How would the required sample size change if the probability of Type I error \( \alpha \) was increased?
2. How do we determine Z-scores for specific probabilities in hypothesis testing?
3. What is the impact of a larger standard deviation on sample size requirements?
4. How would this calculation differ if we were testing \( H_1: \mu < 40 \) instead?
5. What happens to Type II error probability if the sample size is reduced?

### Tip
In hypothesis testing, increasing the sample size reduces the probability of Type II error, increasing the test’s power.

It is desired to test the null hypothesis μ = 40 against the alternative hypothesis μ > 40 on the basis of a random sample from a population with standard deviation 4. If the probability of a Type I error is to be 0.03 and the probability of Type II error is to be 0.08 for μ = 39, find the required size of the sample.

Learn how to calculate the required sample size for hypothesis testing with specific Type I and Type II error probabilities. This example involves testing the null hypothesis μ = 40 against the alternative μ > 40 with a standard deviation of 4, α = 0.03, and β = 0.08.

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