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IAL 2025 Jan Q6

A Level / Edexcel / S1

IAL 2025 Jan Paper · Question 6

题目

Problem

As a part of a selection process, applicants for a television game show have to take a test, and then complete a task as quickly as possible. The test score, ww, and the time taken to complete the task, tt minutes, are recorded for each applicant.

The summary statistics below represent the data for a random sample of 3030 applicants.

Swt= −1648.83,Sww=2396.97,∑w= 839,∑t=635,∑t2= 14 837.\begin{align*} S_{wt}=&\,-1648.83,\qquad S_{ww}=2396.97,\\[2mm] \sum w=&\,839,\qquad \sum t=635,\\[2mm] \sum t^2=&\,14\,837. \end{align*}

(a) Show that the product moment correlation coefficient for these data is −0.901-0.901 to 33 significant figures.

(2)

A scatter diagram of tt against ww is plotted for these data.

(b) State two features of the graph that you would expect to see, given the correlation coefficient in part (a)

(2)

(c) Calculate the equation of the regression line of tt on ww in the form

t=a+bwt=a+bw

Give the values of the constants aa and bb to 33 significant figures.

(4)

(d) Give an interpretation of the gradient of this regression line.

(1)

The test score, ww, was a score out of 5050. The manager of the selection process now decides to double all the values of ww to make them into percentages. The manager then recalculates the product moment correlation coefficient and the equation of the regression line.

(e) State, for each of the following, whether the value would increase or decrease or stay the same as a result of applying this change.

(i) The product moment correlation coefficient

(ii) The magnitude of the gradient of the regression line

(iii) The tt intercept of the regression line

(3)

解答

(a)

解法一

思路

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先用 ∑t2\sum t^2、∑t\sum t 和 nn 求 SttS_{tt}，再代入 PMCC 公式

r=SwtSwwStt.\begin{align*} r=\frac{S_{wt}}{\sqrt{S_{ww}S_{tt}}}. \end{align*}

答题过程

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First,

Stt= ∑t2−(∑t)2n= 14837−635230= 1396.166….\begin{align*} S_{tt} =&\,\sum t^2-\frac{(\sum t)^2}{n}\\[3mm] =&\,14837-\frac{635^2}{30}\\[3mm] =&\,1396.166\ldots. \end{align*}

Then

r= SwtSwwStt= −1648.832396.97(1396.166…)= −0.901313….\begin{align*} r =&\,\frac{S_{wt}}{\sqrt{S_{ww}S_{tt}}}\\[3mm] =&\,\frac{-1648.83}{\sqrt{2396.97(1396.166\ldots)}}\\[3mm] =&\,-0.901313\ldots. \end{align*}

Therefore

r=−0.901\begin{align*} r=-0.901 \end{align*}

to 33 significant figures.

(b)

解法一

思路

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rr 接近 −1-1，所以散点应大致贴近一条直线，而且斜率为负。注意这里不能只说 negative correlation，要描述 scatter diagram 的可见特征。

答题过程

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The points would lie close to a straight line.

The line would have a negative gradient, so as the score increases, the time tends to decrease.

(c)

解法一

思路

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回归线 tt on ww 的斜率是

b=SwtSww.\begin{align*} b=\frac{S_{wt}}{S_{ww}}. \end{align*}

截距用

a=tˉ−bwˉ.\begin{align*} a=\bar{t}-b\bar{w}. \end{align*}

答题过程

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The gradient is

b= SwtSww= −1648.832396.97= −0.68788….\begin{align*} b =&\,\frac{S_{wt}}{S_{ww}}\\[3mm] =&\,\frac{-1648.83}{2396.97}\\[3mm] =&\,-0.68788\ldots. \end{align*}

Also,

tˉ=63530,wˉ=83930.\begin{align*} \bar{t}=\frac{635}{30},\qquad \bar{w}=\frac{839}{30}. \end{align*}

So

a= tˉ−bwˉ= 63530−(−0.68788…)83930= 40.404….\begin{align*} a =&\,\bar{t}-b\bar{w}\\[3mm] =&\,\frac{635}{30} -(-0.68788\ldots)\frac{839}{30}\\[3mm] =&\,40.404\ldots. \end{align*}

Therefore the regression line is

t=40.4−0.688w.\begin{align*} t=40.4-0.688w. \end{align*}

(d)

解法一

思路

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斜率是 −0.688-0.688，意思是 test score 每增加 11 分，预测完成任务的时间平均减少 0.6880.688 分钟。

答题过程

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For each increase of 11 in the test score, the time taken is expected to decrease by about 0.6880.688 minutes on average.

(e)

解法一

思路

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把 ww 全部乘以 22 不会改变散点的线性强弱，所以 PMCC 不变。横轴数值变成原来的两倍，同样的 tt 变化对应更大的 ww 变化，所以斜率的大小会减小。w=0w=0 的位置没有改变，所以 tt 截距不变。

答题过程

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(i) The product moment correlation coefficient stays the same.

(ii) The magnitude of the gradient decreases.

(iii) The tt intercept stays the same.