因为样本量＞30

问题如下：

A junior risk analyst at an asset management firm is monitoring the performance of a recently launched mutual fund against a benchmark index. The analyst uses the last 36 months of excess returns data to construct a confidence interval that can be used to test the one-sided hypothesis that the average excess monthly return of the mutual fund is greater than 0%. Information about the hypothesis test is given below:

• A 5% significance level is used.
  
• Excess monthly returns are assumed to follow a normal distribution.
• The standard deviation of excess returns over the previous 36 months is 0.05.
• The average excess monthly return over the previous 36 months was 1.2%.

Which of the following provides the correct confidence interval to be used as the decision criterion for the hypothesis test?

选项：

A.

(−∞, −0.17%]

B. [−0.17%, ∞) C. [−0.17%, 2.57%] D. [1.20%, 2.57%]

解释：

Explanation: B is correct. For a one-sided test with alternative hypothesis of the form H1: µ > µ0, the confidence interval can be constructed as [µ0 − (Z_α) * (σ/√n), ∞] = [1.2% − (1.645) * (.05/√36), ∞] = [−0.17%, ∞]

A is incorrect. This is close to the confidence interval associated with the alternative hypothesis H1: µ < µ0, given by [−∞, 1.2% + 1.645 * (.05/√36)] where 1.645 is the corresponding critical value for a normal distribution with alpha = .05 on one side.

C is incorrect. The alternative hypothesis calls for a one-sided test; this confidence interval is the result when a two-sided test is conducted.

D is incorrect. The alternative hypothesis calls for a one-sided test. Furthermore, this results from the confidence interval [µ, µ + (Z_α) * (σ/√n)] which is not used in any hypothesis test.

Learning Objective: Understand how a hypothesis test and a confidence interval are related.

Reference: Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson, 2023, Chapter 6, Hypothesis Testing [QA-6].