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IAL 2022 Oct Q3

A Level / Edexcel / S1

IAL 2022 Oct Paper · Question 3

题目

Problem

Morgan is investigating the body length, bb centimetres, of squirrels. A random sample of 88 squirrels is taken and the data for each squirrel is coded using

x=b−212x=\frac{b-21}{2}

The results for the coded data are summarised below

∑x=−1.2∑x2=5.1\sum x=-1.2\qquad \sum x^2=5.1

(a) Find the mean of bb

(3)

(b) Find the standard deviation of bb

(3)

A 99th squirrel is added to the sample. Given that for all 99 squirrels ∑x=0\sum x=0

(c) find

(i) the body length of the 99th squirrel,

(ii) the standard deviation of xx for all 99 squirrels.

(4)

解答

(a)

解法一

思路

展开

由 x=b−212x=\dfrac{b-21}{2} 得 b=2x+21b=2x+21，所以 bˉ=2xˉ+21\bar{b}=2\bar{x}+21。

答题过程

展开 xˉ=−1.28=−0.15.\begin{align*} \bar{x}=\frac{-1.2}{8}=-0.15. \end{align*}

Since

b=2x+21,\begin{align*} b=2x+21, \end{align*}

we have

bˉ=2(−0.15)+21=20.7.\begin{align*} \bar{b}=2(-0.15)+21=20.7. \end{align*}

The mean body length is

20.7 cm.\begin{align*} 20.7\text{ cm}. \end{align*}

(b)

解法一

思路

展开

先求 coded variable xx 的 standard deviation。由于 b=2x+21b=2x+21，标准差会乘以 22。

答题过程

展开 σx2= 5.18−(−1.28)2= 0.615.\begin{align*} \sigma_x^2 =&\,\frac{5.1}{8}-\left(\frac{-1.2}{8}\right)^2\\[3mm] =&\,0.615. \end{align*}

So

σx=0.615=0.7842…\begin{align*} \sigma_x=\sqrt{0.615}=0.7842\ldots \end{align*}

Since b=2x+21b=2x+21,

σb=2σx=2(0.7842…)=1.568…\begin{align*} \sigma_b=2\sigma_x=2(0.7842\ldots)=1.568\ldots \end{align*}

The standard deviation of bb is

1.57 cm\begin{align*} 1.57\text{ cm} \end{align*}

to 33 significant figures.

(c)(i)

解法一

思路

展开

原来 88 只 squirrel 的 ∑x=−1.2\sum x=-1.2。加入第 99 只后总和为 00，所以第 99 只的 coded value 是 1.21.2。

答题过程

展开

Let the coded value for the 99th squirrel be x9x_9.

−1.2+x9=0\begin{align*} -1.2+x_9=0 \end{align*}

so

x9=1.2.\begin{align*} x_9=1.2. \end{align*}

Hence

b9=2(1.2)+21=23.4.\begin{align*} b_9=2(1.2)+21=23.4. \end{align*}

The body length is

23.4 cm.\begin{align*} 23.4\text{ cm}. \end{align*}

(c)(ii)

解法一

思路

展开

更新 ∑x2\sum x^2，然后用 n=9n=9 和新的 ∑x=0\sum x=0 求 standard deviation。

答题过程

展开

For all 99 squirrels,

∑x2=5.1+1.22=6.54.\begin{align*} \sum x^2=5.1+1.2^2=6.54. \end{align*}

Also,

∑x=0.\begin{align*} \sum x=0. \end{align*}

Therefore

σx= 6.549−02= 0.852…\begin{align*} \sigma_x =&\,\sqrt{\frac{6.54}{9}-0^2}\\[3mm] =&\,0.852\ldots \end{align*}

So the standard deviation of xx is

0.852\begin{align*} 0.852 \end{align*}

to 33 significant figures.