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IAL 2024 June S3 Q2

A Level / Edexcel / S3

IAL 2024 June Paper · Question 2

题目

Problem

Aarush is asked to estimate the price of 7 kettles and rank them in order of decreasing price.

Aarush’s order of decreasing price is D A F C B G E The actual prices of the 7 kettles are shown in the table below.

KettleABCDEFG
Price (£)99.9914.9934.9749.9919.9729.998.99

(a) Calculate Spearman’s rank correlation coefficient between Aarush’s order and the actual order. Use a rank of 1 for the highest priced kettle. Show your working clearly.

(4)

(b) Using a 5% level of significance, test whether or not there is evidence to suggest that Aarush is able to rank kettles in order of decreasing price. You should state your hypotheses and critical value.

(4)

(c) Explain why Aarush did not use the product moment correlation coefficient in this situation.

(1)

Aarush discovered that kettle A’s price was recorded incorrectly and should have been £49.99 rather than £99.99.

(d) Explain what effect this has on the rankings for the price.

(1)
(Total for Question 2 is 10 marks)
题目中文翻译

Aarush 被要求估计 7 个电水壶的价格，并按价格从高到低排序。

Aarush 按价格从高到低的顺序为 D A F C B G E。 7 个电水壶的实际价格如下表所示。

电水壶ABCDEFG
价格 (£)99.9914.9934.9749.9919.9729.998.99

(a) 计算 Aarush 的排序与实际排序之间的 Spearman 等级相关系数。 对最高价格的电水壶记为 1 号名次。 请清楚写出你的过程。

(b) 在 5% 显著性水平下，检验是否有证据表明 Aarush 能按价格从高到低对电水壶排序。 你应写出原假设和临界值。

(c) 解释为什么 Aarush 在这种情况下没有使用积差相关系数。

Aarush 发现电水壶 A 的价格记录有误，应为 £49.99，而不是 £99.99。

(d) 解释这对价格排名有什么影响。

解答

(a)

解法一

思路

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先把 Aarush 给出的次序和实际价格都换成名次，其中最高价为 1。逐个计算名次差 dd，再将 ∑d2\sum d^2 代入无并列名次的 Spearman 公式。

答题过程

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The ranks are

KettleABCDEFG
Aarush’s rank2541736
Actual rank1632547
dd1−1-11−1-12−1-1−1-1
d2d^21111411

Therefore,

∑d2=1+1+1+1+4+1+1=10.\sum d^2=1+1+1+1+4+1+1=10.

Hence

rs= 1−6∑d2n(n2−1)= 1−6(10)7(72−1)= 2328= 0.821(3 s.f.).\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(10)}{7(7^2-1)}\\ =&\,\frac{23}{28}\\ =&\,\boxed{0.821}\quad\text{(3 s.f.)}. \end{align*}

(b)

解法一

思路

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“能够正确排序”对应正等级相关，因此使用单尾检验。把 (a) 的样本 Spearman 系数与 n=7n=7、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 Spearman rank correlation coefficient. The hypotheses are

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

For n=7n=7, the critical value for a one-tailed test at the 5%5\% significance level is 0.71430.7143. The rejection region is

rs⩾0.7143.r_s\geqslant0.7143.

Since 0.821>0.71430.821>0.7143, H0H_0 is rejected. There is evidence at the 5%5\% significance level that Aarush is able to rank the kettles in order of decreasing price.

(c)

解法一

思路

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积差相关系数要求成对的数值数据，但 Aarush 只给出了次序，没有给出他的价格估计值；他的资料是等级数据。

答题过程

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Aarush gave only ranks, rather than numerical estimates of the prices, so product moment correlation is not appropriate.

(d)

解法一

思路

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修正后 A 与 D 的价格同为 £49.99，并列占据实际排名的第 1、2 位。并列名次取这两个位置的平均值，因此两者都排 1.5。

答题过程

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Kettles A and D would be tied for first and second place. Therefore, each would be assigned the average rank

1+22=1.5.\frac{1+2}{2}=\boxed{1.5}.