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, $w$ , and the time taken to complete the task, $t$ minutes, are recorded for each applicant.

The summary statistics below represent the data for a random sample of $30$ applicants.

\begin{align*} S_{wt}=&\,-1648.83,\\ S_{ww}=&\,2396.97,\\ \sum w=&\,839,\\ \sum t=&\,635,\\ \sum t^2=&\,14\,837. \end{align*}

(a) Show that the product moment correlation coefficient for these data is $-0.901$ to $3$ significant figures.

(2)

A scatter diagram of $t$ against $w$ 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)

t=a+bw

Give the values of the constants $a$ and $b$ to $3$ significant figures.

(4)

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

(1)

The test score, $w$ , was a score out of $50$ . The manager of the selection process now decides to double all the values of $w$ 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 $t$ intercept of the regression line

(3)

题目中文翻译

电视游戏节目的选拔中，申请者先参加测试，再尽快完成一项任务。记录每位申请者的测试分数 $w$ 和完成任务的时间 $t$ 分钟。

随机抽取 $30$ 位申请者，汇总统计量为：

\begin{align*} S_{wt}=&\,-1648.83,\\ S_{ww}=&\,2396.97,\\ \sum w=&\,839,\\ \sum t=&\,635,\\ \sum t^2=&\,14837. \end{align*}

(a) 证明这些数据的积矩相关系数为 $-0.901$ ，精确到 3 位有效数字。（2 分）

用这些数据绘制 $t$ 对 $w$ 的散点图。

(b) 根据 (a) 的相关系数，写出预期在图上看到的两个特征。（2 分）

(d) 解释这条回归直线斜率的含义。（1 分）

测试满分为 $50$ 分。选拔负责人决定将全部 $w$ 值乘以 $2$ ，转换为百分数，再重新计算积矩相关系数和回归直线方程。

(e) 写出下列各值在作出此变化后会增大、减小还是保持不变：

(i) 积矩相关系数；(ii) 回归直线斜率的绝对值；(iii) 回归直线在 $t$ 轴上的截距。（共 3 分）

解答

(a)

解法一

思路

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先用 $\sum t^2$ 、 $\sum t$ 和 $n$ 求 $S_{tt}$ ，再代入 PMCC 公式

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

答题过程

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First,

\begin{align*} S_{tt} =&\,\sum t^2-\frac{(\sum t)^2}{n}\\ =&\,14837-\frac{635^2}{30}\\ =&\,1396.166\ldots. \end{align*}

Then

\begin{align*} r =&\,\frac{S_{wt}}{\sqrt{S_{ww}S_{tt}}}\\ =&\,\frac{-1648.83}{\sqrt{2396.97(1396.166\ldots)}}\\ =&\,-0.901313\ldots. \end{align*}

Therefore

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

to $3$ significant figures.

(b)

解法一

思路

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$r$ 接近 $-1$ ，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 $t$ on $w$ 的斜率是

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

截距用

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

答题过程

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The gradient is

\begin{align*} b =&\,\frac{S_{wt}}{S_{ww}}\\ =&\,\frac{-1648.83}{2396.97}\\ =&\,-0.68788\ldots. \end{align*}

Also,

\begin{align*} \bar{t}=&\,\frac{635}{30},\\ \bar{w}=&\,\frac{839}{30}. \end{align*}

\begin{align*} a =&\,\bar{t}-b\bar{w}\\ =&\,\frac{635}{30} -(-0.68788\ldots)\frac{839}{30}\\ =&\,40.404\ldots. \end{align*}

Therefore the regression line is

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

(d)

解法一

思路

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斜率是 $-0.688$ ，意思是 test score 每增加 $1$ 分，预测完成任务的时间平均减少 $0.688$ 分钟。

答题过程

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For each increase of $1$ in the test score, the time taken is expected to decrease by about $0.688$ minutes on average.

(e)

解法一

思路

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把 $w$ 全部乘以 $2$ 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 $t$ 变化对应更大的 $w$ 变化，所以斜率的大小会减小。 $w=0$ 的位置没有改变，所以 $t$ 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 $t$ intercept stays the same.

S1 2025 Jan Q6

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

思路

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(e)

解法一

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