R9 练习: Parametric and Non-Parametric Tests

R9 练习: 参数与非参数检验

考纲范围

• Compare and contrast parametric and nonparametric tests, and describe situations where each is the more appropriate type of test.

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Q1.

An analyst wonders whether the mean monthly returns of two independent 2-year portfolios are equal. The distributions for the two portfolios’ returns are unknown. Which of the following tests can the analyst use?

A. t-test.

B. Non-parametric test.

C. Paired comparisons test.

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答案：B

解析：当总体分布未知时，参数检验（如t检验）的前提假设不满足。

选项	判断	解析
A	✗	t检验要求总体服从正态分布或样本量足够大。2年期组合只有24个月度数据，分布未知时不宜使用t检验
B	✓	非参数检验不需要对总体分布做假设，适用于分布未知的情况
C	✗	配对检验用于配对数据，且同样需要分布假设

关联：R9: Parametric and Non-Parametric Tests

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Q2.

Compared with a parametric test, a non-parametric test:

A. concerns things other than parameter values.

B. requires a known population variance.

C. cannot be used if data are given in ranks.

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答案：A

解析：非参数检验与参数检验的区别。

选项	判断	解析
A	✓	非参数检验关注的不是参数值（如均值、方差），而是分布的形状、中位数、独立性等非参数特征
B	✗	非参数检验不需要已知总体方差，这恰恰是参数检验的要求
C	✗	非参数检验恰好可以处理排序（rank）数据，这是其优势之一

关联：R9: Parametric and Non-Parametric Tests

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Q3.

The correlation coefficient is most often used in the analysis of public companies or asset classes. A portfolio manager is analyzing how the return on the bond market is related to that on the stock market. He has computed the sample correlation between the returns of the two markets to be -0.54 with 260 weekly data. With the 0.01 significance level, the pairwise correlation between the returns of these two markets is (Hint: With 258 degrees of freedom and a two-sided test with a 1% level of significance, the critical value is +/-2.595.):

A. significantly different from zero.

B. no difference with zero.

C. lack of evidence to conclude.

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答案：A

解析：检验相关系数是否显著不为零。

计算过程： $t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}=\frac{-0.54\sqrt{258}}{\sqrt{1-0.2916}}=\frac{-0.54\times 16.062}{\sqrt{0.7084}}=\frac{-8.673}{0.8417}=-10.30$

|t| = 10.30 > 2.595（临界值），拒绝H₀: ρ = 0。

选项	判断	解析
A	✓
B	✗	检验统计量远超临界值，应拒绝零假设
C	✗	证据充分

关联：R9: Parametric and Non-Parametric Tests

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Q4.

Which of the following statements is least likely correct regarding the Spearman rank correlation coefficient?

A. The Spearman rank correlation coefficient is calculated on the ranks of the two variables within their respective samples.

B. With n as the sample size and $d_i$ as the difference between the ranks for each pair of observations, the Spearman rank correlation coefficient is given by: $r_s=1-\frac{6{\sum }_{i=1}^n{d}_i}{n(n^2-1)}$

C. The hypotheses are $H_0$: $r_s=0$ and $H_a$: $r_s\ne 0$.

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答案：B

解析：Spearman秩相关系数的公式和性质。

选项	判断	解析
A	✓	正确，Spearman秩相关是基于两个变量的秩次计算的
B	✗	公式中应该是 $d_i^2$（差异的平方），不是 $d_i$。正确公式：$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$
C	✓	正确，假设检验为双尾检验

关联：R9: Parametric and Non-Parametric Tests

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Q5.

Assume an analyst is evaluating the correlation between the returns of two investment funds, A and B, which have comparable risk levels over a 35-year period. Due to concerns that the assumptions for parametric correlation may not be satisfied, the analyst attempts to conduct a hypothesis test using Spearman rank correlations at a 5% significance level. Which of the following statements is least likely correct:

A. The hypotheses are $H_0$: $r_s=0$ and $H_a$: $r_s\ne 0$.

B. Assuming the Spearman rank correlation is 0.65, the calculated test statistic is closest to 4.91.

C. For a two-tailed test, given the critical values for 33 degrees of freedom are +/-2.0345, the analyst fails to reject the null hypothesis that the Spearman rank correlation coefficient is zero.

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答案：C

解析：Spearman秩相关的假设检验。

计算过程： $t=\frac{r_s\sqrt{n-2}}{\sqrt{1-r_s^2}}=\frac{0.65\sqrt{33}}{\sqrt{1-0.4225}}=\frac{0.65\times 5.745}{\sqrt{0.5775}}=\frac{3.734}{0.7599}=4.914$

|t| = 4.914 > 2.0345，应该拒绝H₀。

选项	判断	解析
A	✓	假设设定正确
B	✓	t ≈ 4.91，计算正确
C	✗	不正确。t = 4.91 > 2.0345，应该拒绝H₀（不是”fails to reject”）

关联：R9: Parametric and Non-Parametric Tests

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Q6.

Which of the following statements about the chi-square test of the independence of data classification in a contingency table is incorrect?

A. It is a nonparametric test used to test for the independence of two categorical variables.

B. The rejection region is on the left tail.

C. The degrees of freedom of the chi-square test statistic is calculated as (row number - 1) x (column number - 1).

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答案：B

解析：列联表卡方独立性检验的特征。

选项	判断	解析
A	✓	正确，卡方独立性检验是非参数检验，用于检验两个分类变量的独立性
B	✗	不正确。卡方检验的拒绝域在右尾（right tail），不是左尾
C	✓	正确，自由度 = (行数-1) x (列数-1)

关联：R9: Parametric and Non-Parametric Tests

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Q7.

To test whether an exchange-traded fund (ETF)‘s size is independent of its investment type based on a size-type contingency table, an analyst decides to conduct a hypothesis test. Which statement is(are) least likely correct regarding the test?

Statement I: The analyst should perform a test of independence using a nonparametric test. Statement II: The appropriate test statistic should be t-distributed.

A. Statement I.

B. Statement II.

C. Both statement I and statement II.

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答案：B

解析：列联表独立性检验的方法选择。

选项	判断	解析
Statement I	✓	正确，列联表独立性检验使用非参数检验（卡方检验）
Statement II	✗	不正确，适当的检验统计量应服从卡方分布，不是t分布

只有Statement II不正确，答案为B。

关联：R9: Parametric and Non-Parametric Tests