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S3 2021 June Q1

A Level / Edexcel / S3

IAL 2021 June Paper · Question 1

题目

Problem

1.

A plant biologist claims that as the percentage moisture content of the soil in a field increases, so does the percentage plant coverage. He splits the field into equal areas labelled A, B, C, D and E and measures the percentage plant coverage and the percentage moisture content for each area. The results are shown in the table below.

ABCDE
Coverage %10122506
Moisture %3020401025

(a) Calculate Spearman’s rank correlation coefficient for these data.

(5)

(b) Stating your hypotheses clearly, test at the 5% level of significance, whether or not these data provide support for the plant biologist’s claim.

(4)
(Total for Question 1 is 9 marks)
题目中文翻译

一位植物学家声称，随着田地土壤含水百分比的增加，植物覆盖百分比也会增加。他把这块田地分成大小相等的 A、B、C、D、E 五个区域，并测量每个区域的植物覆盖百分比和土壤含水百分比。结果如下表。

ABCDE
覆盖率 %10122506
含水率 %3020401025

(a) 求这些数据的 Spearman 秩相关系数。

(5)

(b) 清楚写出假设，在 5% 显著性水平下检验：这些数据是否支持这位植物学家的说法。

(4)

解答

(a)

解法一

思路

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分别把植物覆盖率和土壤含水率从大到小排列并赋予名次，再按区域配对。求出名次差 dd 及 ∑d2\sum d^2 后，代入无并列名次时的 Spearman 秩相关系数公式。

答题过程

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Ranking both variables from highest to lowest gives

AreaABCDE
Coverage rank32154
Moisture rank24153
dd1−2-2001
d2d^214001

Therefore,

∑d2=6.\sum d^2=6.

With n=5n=5 and no tied ranks,

rs= 1−6∑d2n(n2−1)= 1−6(6)5(52−1)= 0.7.\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(6)}{5(5^2-1)}\\ =&\,\boxed{0.7}. \end{align*}

(b)

解法一

思路

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生物学家的主张是含水率越高，覆盖率也越高，即正相关，因此进行单尾 Spearman 秩相关检验。把 (a) 的 rsr_s 与 n=5n=5、5%5\% 单尾检验的临界值比较。

答题过程

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Let ρs\rho_s be the population rank correlation coefficient between percentage moisture content and percentage plant coverage.

H0: ρs=0,H1: ρs>0.\begin{align*} H_0:&\ \rho_s=0,\\ H_1:&\ \rho_s>0. \end{align*}

For n=5n=5, the critical value for a one-tailed test at the 5%5\% significance level is 0.90.9.

Since

rs=0.7<0.9,r_s=0.7<0.9,

the test statistic does not lie in the critical region, so H0H_0 is not rejected. There is insufficient evidence at the 5%5\% significance level to support the plant biologist’s claim that percentage plant coverage increases as percentage soil moisture content increases.