题目

Problem

Chi wanted to summarise the scores of the 39 competitors in a village quiz. He started to produce the following stem and leaf diagram

He did not complete the stem and leaf diagram but instead produced the following box plot.

Chi defined an outlier as a value that is greater than $Q_3+1.5(Q_3-Q_1)$ or less than $Q_1-1.5(Q_3-Q_1)$

(a) Find (i) the interquartile range (ii) the range.

(2)

(b) Describe, giving a reason, the skewness of the distribution of scores.

(2)

Albert and Beth asked for their scores to be checked.

Albert’s score was changed from 25 to 37

Beth’s score was changed from 54 to 60

(7)

Some of the competitors complained that the questions were biased towards the younger generation. The product moment correlation coefficient between the age of the competitors and their score in the quiz is $-0.187$

(d) State, giving a reason, whether or not the complaint is supported by this statistic.

(2)

题目中文翻译

Chi 想汇总一次村庄知识竞赛中 39 名参赛者的分数。他开始制作题图中的茎叶图，但没有完成，转而制作了箱线图。

Chi 把大于 $Q_3+1.5(Q_3-Q_1)$ 或小于 $Q_1-1.5(Q_3-Q_1)$ 的数值定义为离群值。

(a) 求：(i) 四分位距；(ii) 极差。

(b) 描述分数分布的偏态，并说明理由。

Albert 与 Beth 要求核对分数。Albert 的分数由 25 改为 37，Beth 的分数由 54 改为 60。

部分参赛者投诉题目偏向年轻一代。参赛者年龄与竞赛分数的积矩相关系数为 $-0.187$ 。

(d) 说明这一统计量是否支持该投诉，并给出理由。

解答

(a)

(i)

思路

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从箱线图读出 $Q_1=33$ 与 $Q_3=47$ 。

答题过程

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\begin{align*} \operatorname{IQR}=47-33=14. \end{align*}

(ii)

思路

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从图中读出最小值 11 与最大值 54。

答题过程

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The range is

\begin{align*} 54-11=43. \end{align*}

(b)

解法一

思路

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比较中位数到上下四分位数的距离。左边距离 $42-33=9$ ，右边距离 $47-42=5$ ，左侧更长，所以负偏态。

答题过程

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\begin{align*} Q_2-Q_1=42-33=9, \end{align*}

whereas

\begin{align*} Q_3-Q_2=47-42=5. \end{align*}

Since $Q_2-Q_1>Q_3-Q_2$ , the distribution is negatively skewed.

(c)

解法一

思路

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改分后，下四分位数变成 35，上四分位数仍为 47，中位数仍为 42。重新计算 IQR 和离群值边界。

答题过程

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The new lower quartile is

\begin{align*} Q_1=35. \end{align*}

The median remains $42$ , and the upper quartile remains $47$ .

So the new IQR is

\begin{align*} 47-35=12. \end{align*}

The lower outlier boundary is

\begin{align*} 35-1.5(12)=17. \end{align*}

The upper outlier boundary is

\begin{align*} 47+1.5(12)=65. \end{align*}

So the outliers are $11$ and $15$ .

The updated box plot should have a box from $35$ to $47$ , median $42$ , whiskers to $18$ and $60$ , and outliers at $11$ and $15$ .

(d)

解法一

思路

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$-0.187$ 很接近 0，说明相关性很弱，不能支持“年龄越小分数越高”这种说法。

答题过程

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The complaint is not supported.

The product moment correlation coefficient $-0.187$ is close to 0, so there is only very weak correlation between age and score.

S1 2019 June Q2

题目

解答

(a)

(i)

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(ii)

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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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