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S3 2023 June Q1

A Level / Edexcel / S3

IAL 2023 June Paper · Question 1

题目

Problem

1.

(a) State two conditions under which it might be more appropriate to use Spearman’s rank correlation coefficient rather than the product moment correlation coefficient.

(2)

A random sample of 10 melons was taken from a market stall. The length, in centimetres, and maximum diameter, in centimetres, of each melon were recorded. The Spearman’s rank correlation coefficient between the results was −0.673-0.673

(b) Test, at the 5% level of significance, whether or not there is evidence of a correlation. State clearly your hypotheses and the critical value used.

(4)

The product moment correlation coefficient between the results was −0.525-0.525

(c) Test, at the 5% level of significance, whether or not there is evidence of a negative correlation. State clearly your hypotheses and the critical value used.

(3)
(Total for Question 1 is 9 marks)
题目中文翻译

(a) 写出在什么情况下，使用 Spearman 秩相关系数可能比使用积差相关系数更合适的两个条件。

(2)

从一个市场摊位抽取了 10 个瓜的随机样本。记录了每个瓜的长度（厘米）和最大直径（厘米）。 结果的 Spearman 秩相关系数为 −0.673-0.673。

(b) 在 5% 显著性水平下，检验是否有相关性的证据。请清楚写出原假设和所用临界值。

(4)

结果的积差相关系数为 −0.525-0.525。

(c) 在 5% 显著性水平下，检验是否有负相关的证据。 请清楚写出原假设和所用临界值。

(3)

解答

(a)

解法一

思路

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Spearman 秩相关系数只使用数据的次序，不要求变量之间呈线性关系。因此，当数据本身是顺序数据，或预计变量之间存在单调但非线性的关系时，它通常比积差相关系数更合适。

答题过程

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It may be more appropriate to use Spearman’s rank correlation coefficient when:

  1. the data are ordinal, for example judges’ rankings;
  2. a non-linear relationship between the variables is expected.

(b)

解法一

思路

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题目问“是否存在相关性”，没有预先指定正相关或负相关，因此使用双尾 Spearman 秩相关检验。把样本系数 −0.673-0.673 与 n=10n=10、5%5\% 双尾临界值比较。

答题过程

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Let ρ\rho be the population rank correlation coefficient between the length and maximum diameter of the melons.

H0: ρ=0,H1: ρ≠0.\begin{align*} H_0:&\ \rho=0,\\ H_1:&\ \rho\ne0. \end{align*}

For n=10n=10 at the 5%5\% significance level in a two-tailed test, the critical values are

rs=±0.6485.r_s=\pm0.6485.

The observed value is

rs=−0.673<−0.6485,r_s=-0.673<-0.6485,

so it lies in the critical region. Therefore, H0H_0 is rejected. There is sufficient evidence at the 5%5\% significance level of a correlation between the length and maximum diameter of the melons.

(c)

解法一

思路

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这次明确检验负相关，所以使用左尾积差相关检验。将样本积差相关系数 −0.525-0.525 与 n=10n=10、5%5\% 单尾临界值比较。

答题过程

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Let ρ\rho be the population product moment correlation coefficient between the length and maximum diameter of the melons.

H0: ρ=0,H1: ρ<0.\begin{align*} H_0:&\ \rho=0,\\ H_1:&\ \rho<0. \end{align*}

For n=10n=10 at the 5%5\% significance level in a one-tailed test, the critical value is

r=−0.5494.r=-0.5494.

Since

r=−0.525>−0.5494,r=-0.525>-0.5494,

the observed value does not lie in the critical region. Therefore, H0H_0 is not rejected. There is insufficient evidence at the 5%5\% significance level of a negative correlation between the length and maximum diameter of the melons.