R8 练习: Hypothesis Testing

R8 练习: 假设检验

考纲范围

• Explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test.
• Construct hypothesis tests and determine their statistical significance, the associated Type I and Type II errors, and power of the test given a significance level.

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Q1.

An analyst believes that the inflation-adjusted mean return on T-bills with a maturity of 3 months is less than zero. In the hypothesis testing, the null and alternative hypotheses should be stated as:

A. $H_0$: $\mu \ge 0$ versus $H_a$: $\mu <0$.

B. $H_0$: $\mu =0$ versus $H_a$: $\mu \ne 0$.

C. $H_0$: $\mu \le 0$ versus $H_a$: $\mu >0$.

查看答案与解析

答案：A

解析：分析师认为均值小于零，这是他想要证明的（备择假设）。原假设总是包含等号。

选项	判断	解析
A	✓	正确。分析师想证明μ<0，所以Ha: μ<0，对应的H₀: μ≥0（单尾左侧检验）
B	✗	这是双尾检验，不符合分析师”小于零”的方向性假设
C	✗	方向反了，这是检验μ>0的单尾右侧检验

关联：R8: Hypothesis Testing

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Q2.

Two analysts (A and B) believe company XYZ’s monthly return is more than 4%, thus they intend to do the hypotheses test that H₀: $\mu \le 4\%$ versus H_a: $\mu >4\%$. Analyst A and Analyst B conduct the test at the significance level of 5% and 10%, respectively. The p-value calculated by the computer is 0.065. Who should reject the null hypothesis?

A. Analyst A.

B. Analyst B.

C. Both of them.

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答案：B

解析：当p值小于显著性水平时，拒绝原假设。

计算过程：

• Analyst A: α = 5% = 0.05, p-value = 0.065 > 0.05 → 不拒绝H₀
• Analyst B: α = 10% = 0.10, p-value = 0.065 < 0.10 → 拒绝H₀

选项	判断	解析
A	✗	p-value(0.065) > α(0.05)，Analyst A不能拒绝H₀
B	✓	p-value(0.065) < α(0.10)，只有Analyst B可以拒绝H₀
C	✗	Analyst A不能拒绝

关联：R8: Hypothesis Testing

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Q3.

Which of the following statements is most likely to be correct?

A. Type I error describes a situation in which a true null hypothesis has been incorrectly rejected.

B. Type II error describes a situation in which a true null hypothesis has not been rejected.

C. Power of the test can be expressed as 1 minus the probability of Type I error.

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答案：A

解析：假设检验中的错误类型和检验力。

选项	判断	解析
A	✓	第一类错误（Type I error）= 原假设为真时错误地拒绝了原假设（“弃真”）
B	✗	Type II error是原假设为假时未能拒绝原假设（“取伪”），不是”true null hypothesis has not been rejected”
C	✗	检验力（Power）= 1 - P(Type II error)，不是1 - P(Type I error)

关联：R8: Hypothesis Testing

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Q4.

During the past 5 years, company A’s annual profit has an average of 25 million and a standard deviation of 5 million. An analyst wants to verify that company A’s actual average annual profit doesn’t exceed 23 million. Assuming the annual profit is normally distributed and the variance is known, which of the following is correct given a 0.05 significance level?

A. He can reject the null hypothesis.

B. He can’t reject the null hypothesis.

C. Information isn’t enough.

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答案：B

解析：分析师想验证均值不超过23百万。

计算过程：

• H₀: μ ≤ 23 versus Ha: μ > 23（分析师想证明利润超过23M）

等等，题意是”verify that actual average doesn’t exceed 23 million”，即H₀: μ ≥ 23, Ha: μ < 23。但样本均值25 > 23，样本证据方向不支持Ha。

实际上：H₀: μ ≤ 23, Ha: μ > 23 $z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}=\frac{25-23}{5/\sqrt{5}}=\frac{2}{2.236}=0.894$

在5%显著性水平下（单尾），临界值z₀.₀₅ = 1.645。z = 0.894 < 1.645，不能拒绝H₀。

选项	判断	解析
A	✗	检验统计量未超过临界值
B	✓	z = 0.894 < 1.645，不能拒绝原假设
C	✗	信息充足

关联：R8: Hypothesis Testing

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Q5.

To test the difference between the mean monthly income growth rate of a Japanese household and that of an Australian household over the past 50 years, assuming they are both independent and normally distributed, which of the following tests is most appropriate?

A. t-test on the difference of means.

B. Paired comparisons test.

C. F-test.

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答案：A

解析：两个独立样本的均值差检验。

选项	判断	解析
A	✓	两个独立且正态分布的样本比较均值差，使用t检验（差异均值t检验）
B	✗	配对检验用于同一对象的前后比较或配对数据，日本和澳洲家庭是独立样本
C	✗	F检验用于比较两个总体方差是否相等，不是比较均值

关联：R8: Hypothesis Testing

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Q6.

When testing whether the variance of a normalized population equals a specific value, which of the following tests is most appropriate?

A. A F-test.

B. A t-test.

C. A chi-square test.

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答案：C

解析：检验单个正态总体的方差是否等于特定值。

选项	判断	解析
A	✗	F检验用于比较两个总体的方差
B	✗	t检验用于检验均值
C	✓	卡方检验（chi-square test）用于检验单个正态总体的方差是否等于特定值

关联：R8: Hypothesis Testing

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Q7.

May, CFA, studies two normally distributed variables X and Y. To test the hypothesis that the variance of X is equal to the variance of Y, she should use a:

A. Chi-square test.

B. F-test.

C. t-test.

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答案：B

解析：比较两个正态总体的方差是否相等。

选项	判断	解析
A	✗	卡方检验用于单个总体方差的检验
B	✓	F检验用于检验两个正态总体方差是否相等
C	✗	t检验用于均值检验

关联：R8: Hypothesis Testing

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Q8.

Which of the following statements regarding hypothesis tests is most likely correct?

A. Chi-square test is used to test the equality of two population variances.

B. Chi-square test has a single degree of freedom (df) n-1 while F-test has two degrees of freedom, namely numerator (df) n1-1 and denominator (df) n2-1.

C. F-test is used to test a single population variance.

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答案：B

解析：各类假设检验的特征比较。

选项	判断	解析
A	✗	卡方检验用于单个总体方差，F检验用于两个总体方差的比较
B	✓	正确。卡方检验df = n-1；F检验有两个自由度：分子df = n₁-1，分母df = n₂-1
C	✗	F检验用于比较两个总体方差，卡方检验用于单个总体方差

关联：R8: Hypothesis Testing