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S1 2019 June Q1

A Level / Edexcel / S1

IAL 2019 June Paper · Question 1

题目

Problem

The heights, xx metres, of 40 children were recorded by a teacher. The results are summarised as follows

∑x=58∑x2=84.829\sum x=58\qquad \sum x^2=84.829

(a) Find the mean and the variance of the heights of these 40 children.

(3)

The teacher decided that these statistics would be more useful in centimetres.

(b) Find

(i) the mean of these heights in centimetres,

(ii) the standard deviation of these heights in centimetres.

(2)

Two more children join the group. Their heights are 130 cm and 160 cm.

(c) (i) State, giving a reason, the mean height of the 42 children.

(ii) Without recalculating the standard deviation, state, giving a reason, whether the standard deviation of the heights of the 42 children will be greater than, less than or the same as the standard deviation of the heights of the group of 40 children.

(4)
题目中文翻译

一位老师记录了 40 名儿童的身高 xx（米），资料汇总如上。

(a) 求这 40 名儿童身高的平均数与方差。

老师认为以厘米表示这些统计量更实用。

(b) 求：(i) 身高的厘米制平均数；(ii) 身高的厘米制标准差。

另有两名儿童加入，他们的身高分别为 130 cm 与 160 cm。

(c) (i) 说明 42 名儿童的平均身高并给出理由；(ii) 不重新计算标准差，说明 42 名儿童身高的标准差比原来 40 人的标准差大、小或相同，并给出理由。

解答

(a)

解法一

思路

展开

直接用 xˉ=∑xn\bar x=\frac{\sum x}{n} 和 σ2=∑x2n−xˉ2\sigma^2=\frac{\sum x^2}{n}-\bar x^2。

答题过程

展开 xˉ=5840=1.45.\begin{align*} \bar x=\frac{58}{40}=1.45. \end{align*}

Also,

σ2=84.82940−(1.45)2=0.018225.\begin{align*} \sigma^2=\frac{84.829}{40}-(1.45)^2=0.018225. \end{align*}

So the mean is 1.451.45 m and the variance is

0.0182.\begin{align*} 0.0182. \end{align*}

(b)

(i)

思路

展开

从米换算成厘米，平均数乘以 100。

答题过程

展开

The mean in centimetres is

1.45×100=145.\begin{align*} 1.45\times100=145. \end{align*}

(ii)

思路

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标准差与原数据使用相同的线性单位，所以也乘以 100。

答题过程

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The standard deviation in metres is

0.018225=0.135….\begin{align*} \sqrt{0.018225}=0.135\ldots. \end{align*}

So the standard deviation in centimetres is

0.135…×100=13.5.\begin{align*} 0.135\ldots\times100=13.5. \end{align*}

(c)

(i)

思路

展开

新加入的两人平均身高恰好等于原平均数，因此整体平均数不变。

答题过程

展开

The mean of the two new heights is

130+1602=145.\begin{align*} \frac{130+160}{2}=145. \end{align*}

This is the same as the original mean, so the mean height of the 42 children is still

145 cm.\begin{align*} 145\text{ cm}. \end{align*}

(ii)

思路

展开

两个新数据距离均值均为 15 cm，大于原标准差 13.5 cm，因此会增加离散程度。

答题过程

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The two new heights are both 1515 cm from the mean, which is more than the original standard deviation of 13.513.5 cm.

Therefore the standard deviation will increase.