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.