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S1 2021 Oct Q2

A Level / Edexcel / S1

IAL 2021 Oct Paper · Question 2

题目

Problem

A large company is analysing how much money it spends on paper in its offices each year. The number of employees in the office, xx, and the amount spent on paper in a year, pp (hundreds),ineachofhundreds), in each of12$ randomly selected offices were recorded.

The results are summarised in the following statistics.

∑x=93Sxx=148.25∑p=273\sum x=93\qquad S_{xx}=148.25\qquad \sum p=273 ∑p2=6602.72∑xp=2347\sum p^2=6602.72\qquad \sum xp=2347

(a) Show that Sxp=231.25S_{xp}=231.25

(1)

(b) Find the product moment correlation coefficient for these data.

(3)

(c) Find the equation of the regression line of pp on xx in the form p=a+bxp=a+bx

(4)

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

(1)

The director of the company wants to reduce the amount spent on paper each year. He wants each office to aim for a model of the form p=45a+12bxp=\dfrac45a+\dfrac12bx, where aa and bb are the values found in part (c).

Using the data for the 9393 employees from the 1212 offices,

(e) estimate the percentage saving in the amount spent on paper each year by the company using the director’s model.

(3)
题目中文翻译

一家大型公司正在分析各办公室每年用于纸张的支出。随机选取 1212 个办公室，记录每个办公室的员工人数 xx 以及一年用于纸张的支出 pp（单位为百美元）。

所得数据用以下统计量概括：

∑x=93Sxx=148.25∑p=273,\sum x=93\qquad S_{xx}=148.25\qquad \sum p=273, ∑p2=6602.72∑xp=2347.\sum p^2=6602.72\qquad \sum xp=2347.

(a) 证明 Sxp=231.25S_{xp}=231.25。

(b) 求这些数据的积矩相关系数。

(c) 求 pp 关于 xx 的回归直线方程，并写成 p=a+bxp=a+bx 的形式。

(d) 解释回归直线斜率的含义。

公司主管希望减少每年的纸张支出。他希望每个办公室采用形如 p=45a+12bxp=\dfrac45a+\dfrac12bx 的模型，其中 aa 和 bb 是 (c) 中求得的值。

利用这 1212 个办公室共 9393 名员工的数据，

(e) 估计采用主管模型后，公司每年纸张支出的节省百分比。

解答

(a)

解法一

思路

展开

用 Sxp=∑xp−∑x∑pnS_{xp}=\sum xp-\dfrac{\sum x\sum p}{n}。

答题过程

展开 Sxp= 2347−93(273)12= 2347−2115.75= 231.25.\begin{align*} S_{xp} =&\,2347-\frac{93(273)}{12}\\ =&\,2347-2115.75\\ =&\,231.25. \end{align*}

(b)

解法一

思路

展开

先求 SppS_{pp}，再代入相关系数公式。

答题过程

展开 Spp= 6602.72−273212= 391.97.\begin{align*} S_{pp} =&\,6602.72-\frac{273^2}{12}\\ =&\,391.97. \end{align*}

Therefore

r= 231.25148.25(391.97)= 0.9593…\begin{align*} r =&\,\frac{231.25}{\sqrt{148.25(391.97)}}\\ =&\,0.9593\ldots \end{align*}

So

r=0.959.\begin{align*} r=0.959. \end{align*}

(c)

解法一

思路

展开

回归线 pp on xx 的斜率是 Sxp/SxxS_{xp}/S_{xx}，截距是 pˉ−bxˉ\bar{p}-b\bar{x}。

答题过程

展开 b= SxpSxx= 231.25148.25= 1.55986…\begin{align*} b =&\,\frac{S_{xp}}{S_{xx}}\\ =&\,\frac{231.25}{148.25}\\ =&\,1.55986\ldots \end{align*}

Also,

a= 27312−1.55986…(9312)= 10.66…\begin{align*} a =&\,\frac{273}{12}-1.55986\ldots\left(\frac{93}{12}\right)\\ =&\,10.66\ldots \end{align*}

Therefore

p=10.7+1.56x.\begin{align*} p=10.7+1.56x. \end{align*}

(d)

解法一

思路

展开

pp 的单位是 hundreds of dollars，所以斜率 1.561.56 表示每多一名员工，纸张花费平均增加 156156 dollars。

答题过程

展开

For each extra employee, the amount spent on paper increases by about 156156 dollars per year on average.

(e)

解法一

思路

展开

先用原数据的平均花费 pˉ=273/12=22.75\bar{p}=273/12=22.75。新模型下代入平均员工数 93/1293/12，再比较节省百分比。

答题过程

展开

The original mean amount spent is

pˉ=27312=22.75.\begin{align*} \bar{p}=\frac{273}{12}=22.75. \end{align*}

Using the director’s model,

new mean p= 45(10.66…)+12(1.55986…)(9312)= 14.573…\begin{align*} \text{new mean }p =&\,\frac45(10.66\ldots)+\frac12(1.55986\ldots)\left(\frac{93}{12}\right)\\ =&\,14.573\ldots \end{align*}

Therefore the percentage saving is

22.75−14.573…22.75×100= 35.94…\begin{align*} \frac{22.75-14.573\ldots}{22.75}\times100 =&\,35.94\ldots \end{align*}

So the estimated saving is about

36%.\begin{align*} 36\%. \end{align*}