Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a \( N(\theta, 1) \) distribution. To test \( H_0: \theta = 0 \) against \( H_1: \theta = 1 \), assume that the critical region is given by \[ \frac{1}{n} \sum_{i=1}^n X_i \geq \frac{3}{4}. \] Then the minimum sample size required so that \( P(\text{Type I error}) \leq 0.05 \) is

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

When testing hypotheses, the Type I error is the probability of rejecting the null hypothesis when it is true. The sample size must be chosen to control the probability of Type I error.

Answer 1

The Correct Option is C

Step 1: Calculate the standard deviation.
For the given problem, we know that \( X_1, X_2, \dots, X_n \) are i.i.d. from a normal distribution with mean \( \theta \) and variance 1. The critical region is given by the sample mean being greater than or equal to \( \frac{3}{4} \). Step 2: Use the Type I error formula.
The Type I error occurs when we reject \( H_0 \) when it is true, so we calculate the sample size such that the probability of making a Type I error is less than or equal to 0.05. Step 3: Conclusion.
After calculating the required sample size, we find that the minimum sample size required is 5. Thus, the correct answer is (C) 5.