题目

Problem

A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams A, B, C, D, E, F, G, H and I. He then ranks the marks for taste and the prices.

The ranks are shown in the table below.

Rank	1	2	3	4	5	6	7	8	9
Price	A	B	E	C	D	F	G	H	I
Taste	A	B	F	E	H	G	I	C	D

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

(5)

(b) Test, at the 5% level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly.

(3)

A friend suggests that it would be better to use the price per 100 grams, $c$ , and the mark for the taste, $m$ , for each strawberry jam rather than rank them.

Given that

S_{cc}=2.0455 \qquad S_{mm}=243.5556 \qquad S_{cm}=16.4943

(c) calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.

(2)

(d) Use your value of the product moment correlation coefficient to test, at the 5% level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.

(3)

(e) State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.

(1)

(Total 14 marks)

题目中文翻译

一家咖啡馆老板想知道草莓果酱的价格是否与果酱的味道有关。他找到一个网站，上面列出了 9 种不同草莓果酱 A、B、C、D、E、F、G、H、I 的每 100 克价格，以及由评审给出的 100 分制味道分数。他随后对味道分数和价格进行了排序。

排序如下表所示。

排名	1	2	3	4	5	6	7	8	9
价格	A	B	E	C	D	F	G	H	I
味道	A	B	F	E	H	G	I	C	D

(a) 求这些数据的 Spearman 秩相关系数。

(b) 在 5% 显著性水平下检验：这些草莓果酱的价格与味道之间是否存在关系。清楚写出假设。

朋友建议，与其做排序，不如直接使用每 100 克价格 $c$ 和味道分数 $m$ 。

已知

S_{cc}=2.0455 \qquad S_{mm}=243.5556 \qquad S_{cm}=16.4943

(d) 使用你求得的积矩相关系数，在 5% 显著性水平下检验：价格与味道分数之间是否有正相关的证据。清楚写出假设。

(e) 说明 (b) 和 (d) 中哪一个检验对咖啡馆老板更合适，并给出理由。

解答

(a)

解法一

思路

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先把题目按名次列出的果酱重新按果酱名称配对，求出每种果酱价格名次与味道名次之差 $d$ 。再计算 $\sum d^2$ ，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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For each jam, let $d$ be its price rank minus its taste rank.

Jam	Price rank	Taste rank	$d$	$d^2$
A	1	1	0	0
B	2	2	0	0
C	4	8	$-4$	16
D	5	9	$-4$	16
E	3	4	$-1$	1
F	6	3	3	9
G	7	6	1	1
H	8	5	3	9
I	9	7	2	4

Therefore,

\sum d^2=56.

There are $n=9$ jams and no tied ranks, so

\begin{align*} r_s =&\,1-\frac{6\sum d^2}{n(n^2-1)}\\ =&\,1-\frac{6(56)}{9(9^2-1)}\\ =&\,\frac{8}{15}\\ =&\,\boxed{0.533}\quad\text{(3 s.f.)}. \end{align*}

(b)

解法一

思路

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题目问价格与味道之间是否存在关系，没有预先指定正相关或负相关，因此必须进行双尾 Spearman 秩相关检验。把 $|r_s|$ 与 $n=9$ 、 $5\%$ 双尾检验的临界值比较。

答题过程

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Let $\rho$ be the population rank correlation coefficient between price and taste.

\begin{aligned} H_0:&\ \rho=0,\\ H_1:&\ \rho\neq0. \end{aligned}

For $n=9$ , the critical value for a two-tailed test at the $5\%$ significance level is $0.700$ .

Since

|r_s|=0.533\ldots<0.700,

$H_0$ is not rejected. There is insufficient evidence at the $5\%$ significance level of a relationship between the price and the taste of these strawberry jams.

(c)

解法一

思路

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积矩相关系数可由题目给出的三个离均差乘积和直接计算，即 $r=S_{cm}/\sqrt{S_{cc}S_{mm}}$ 。

答题过程

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The product moment correlation coefficient is

\begin{align*} r =&\,\frac{S_{cm}}{\sqrt{S_{cc}S_{mm}}}\\ =&\,\frac{16.4943}{\sqrt{2.0455(243.5556)}}\\ =&\,0.7389\ldots\\ =&\,\boxed{0.739}\quad\text{(3 d.p.)}. \end{align*}

(d)

解法一

思路

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这里明确检验是否存在正相关，所以使用单尾 PMCC 检验。把 (c) 的 $r$ 与 $n=9$ 、 $5\%$ 单尾检验的临界值比较，并在结论中同时说明方向及实际变量。

答题过程

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Let $\rho$ be the population product moment correlation coefficient between price and taste mark.

\begin{aligned} H_0:&\ \rho=0,\\ H_1:&\ \rho>0. \end{aligned}

For $n=9$ , the critical value for a one-tailed test at the $5\%$ significance level is $0.5822$ .

Since

r=0.7389\ldots>0.5822,

$H_0$ is rejected. There is sufficient evidence at the $5\%$ significance level of a positive correlation between the price and the taste mark of these strawberry jams.

(e)

解法一

思路

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PMCC 检验要求总体近似服从二元正态分布，并把数值视为有意义的测量尺度；味道分数来自评审的主观判断，这些条件并不可靠。因此应采用只依赖次序的 Spearman 秩检验。

答题过程

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Spearman’s rank correlation test in part (b) is more appropriate because the taste marks are subjective judgements and may not form a meaningful measurement scale. It is therefore not reasonable to assume that price and taste mark have a bivariate normal distribution.

IAL 2021 Oct S3 Q3

题目

解答

(a)

解法一

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(b)

解法一

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(c)

解法一

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(d)

解法一

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(e)

解法一

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