题目

Problem

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

\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.

(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)

解答

(a)

解法一

思路

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直接用 $\bar x=\frac{\sum x}{n}$ 和 $\sigma^2=\frac{\sum x^2}{n}-\bar x^2$ 。

答题过程

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

Also,

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

So the mean is $1.45$ m and the variance is

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

(b)

解法一

思路

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从 metres 转成 centimetres，平均数乘 100，标准差也乘 100。

答题过程

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The mean in centimetres is

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

The standard deviation in metres is

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

So the standard deviation in centimetres is

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

(c)

解法一

思路

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新加入的两个高度是 130 和 160，平均数正好是 145，所以整体平均数不变。它们距离均值都是 15 cm，比原标准差 13.5 cm 大，因此离散程度会增加。

答题过程

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The mean of the two new heights is

\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

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

The two new heights are both $15$ cm from the mean, which is more than the original standard deviation of $13.5$ cm.

Therefore the standard deviation will increase.

IAL 2019 June Q1

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程