The problem involves hypothesis testing for a normal distribution. Here's the breakdown:

Problem Context:

• $X_1, X_2, \dots, X_n$ are random samples from a normal distribution with mean $\mu$ and variance $\sigma^2 = 16$.
• Sample size $n = 16$.
• Significance level $\alpha = 0.05$.
• Null hypothesis: $H_0: \mu = 10$.
• Alternative hypothesis: $H_A: \mu = 15$.
• We are to find the most powerful test (Neyman-Pearson Lemma).

Steps to Solve:

1. Statistical Distribution:

   • Under the null hypothesis $H_0$, the test statistic is: $\bar{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right) = N\left(10, \frac{16}{16}\right) = N(10, 1).$

   • Under $H_A$, $\bar{X} \sim N(15, 1)$.

2. Critical Region:

   • The Neyman-Pearson Lemma tells us that the most powerful test is based on the likelihood ratio: $\frac{f(\bar{X} \mid H_A)}{f(\bar{X} \mid H_0)} > k,$ for some critical value $k$.

3. Finding the Critical Value:

   • We reject $H_0$ if $\bar{X}$ falls in the critical region defined such that the probability of Type I error ($\alpha$) is: $P(\text{Reject } H_0 \mid H_0) = \alpha.$

   • Since $\bar{X} \sim N(10, 1)$ under $H_0$: $P(\bar{X} > c \mid H_0) = 0.05.$

   • From the standard normal distribution, the critical value is: $c = \mu_0 + z_\alpha \cdot \sigma = 10 + 1.645 \cdot 1 = 11.645.$

4. Test Procedure:

   • Reject $H_0$ if $\bar{X} > 11.645$.

5. Power of the Test:

   • To calculate the power, compute: $\beta = P(\text{Reject } H_0 \mid H_A) = P(\bar{X} > 11.645 \mid \mu = 15).$
   • Under $H_A$, $\bar{X} \sim N(15, 1)$: $P(\bar{X} > 11.645) = P\left(Z > \frac{11.645 - 15}{1}\right) = P(Z > -3.355).$
   • From standard normal tables: $P(Z > -3.355) \approx 0.9996.$

Summary:

1. The critical region is $\bar{X} > 11.645$.
2. The power of the test is approximately 0.9996.

Would you like a step-by-step explanation of any specific part?

Related Questions:

1. What is the significance of the Neyman-Pearson Lemma in hypothesis testing?
2. How do you interpret the power of a hypothesis test?
3. What happens if we change the sample size $n$? How does it affect the critical value and power?
4. Can you explain the relationship between $\alpha$ and $\beta$ in hypothesis testing?
5. How would the results change if the variance were unknown?

Tip:

When conducting hypothesis tests, always sketch the sampling distributions under $H_0$ and $H_A$ to visualize the critical regions and probability of errors.