Question

Let \( \{x_1, x_2, \dots, x_n\} \) be a realization of a random sample of size \( n (\geq 2) \) from a \( N(\mu, \sigma^2) \) distribution, where \( -\infty<\mu<\infty \) and \( \sigma>0 \). Which of the following statements is/are true?

• A. P only
• B. Q only
• C. Both P and Q
• D. Neither P nor Q

Hint:

When \( \sigma \) is known, the confidence interval for \( \mu \) is unique. However, when \( \sigma \) is unknown, the confidence interval depends on the sample data and is not unique, as it involves the Student's t-distribution.

Answer 1

The Correct Option is B

We are given a random sample \( \{x_1, x_2, \dots, x_n\} \) from a \( N(\mu, \sigma^2) \) distribution. We need to analyze the two statements \( P \) and \( Q \) about the confidence interval for the population mean \( \mu \).
Step 1: Analyzing statement P.
Statement \( P \) asserts that the 95% confidence interval for \( \mu \) is unique when \( \sigma \) is known. This is true because when \( \sigma \) is known, the confidence interval for \( \mu \) is based on the normal distribution, which is fixed and does not depend on any estimation of \( \sigma \). Therefore, the confidence interval for \( \mu \) is unique when \( \sigma \) is known. However, this is not the main point for the solution.
Step 2: Analyzing statement Q.
Statement \( Q \) states that the 95% confidence interval for \( \mu \) is NOT unique when \( \sigma \) is unknown. This is correct because when \( \sigma \) is unknown, we must estimate it using the sample standard deviation \( S \), which leads to a distribution that depends on the sample data (specifically, the Student's t-distribution). Since the t-distribution depends on the sample size, the confidence interval will vary based on this estimate, and hence it is not unique.
Step 3: Conclusion.
Since statement \( P \) is correct and \( Q \) is also correct, the correct answer is \( Q \) only. Final Answer: \boxed{(B) Q only}