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S1 2019 Oct Q1

A Level / Edexcel / S1

IAL 2019 Oct Paper · Question 1

题目

Problem

Jeremiah is investigating the relationship between the annual heating bill, hh dollars, and the total floor area, ff square metres, of buildings. A random sample of 8 buildings is taken and the data for each building are coded using

x=f−350080andy=h−450080x=\frac{f-3500}{80}\quad\text{and}\quad y=\frac{h-4500}{80}

The results for the coded data are summarised below

∑x=5∑y=0∑xy=1818Sxx=1754\sum x=5\qquad \sum y=0\qquad \sum xy=1818\qquad S_{xx}=1754

(a) Calculate SxyS_{xy}

(1)

(b) Find the equation of the regression line of hh on ff in the form h=a+bfh=a+bf

(6)

(c) Give an interpretation of (i) the value of bb in your regression line, (ii) the value of aa in your regression line.

(2)

(d) Estimate the annual heating bill for a building with a total floor area of 4600 square metres.

(2)
题目中文翻译

Jeremiah 正在研究建筑物每年的供暖费用 hh（美元）与总楼面面积 ff（平方米）之间的关系。他随机抽取 8 栋建筑物，并按题目所示方式把每栋建筑物的数据编码，编码后数据的汇总量如题。

(a) 计算 SxyS_{xy}。

(b) 求 hh 关于 ff 的回归直线，答案写成 h=a+bfh=a+bf 的形式。

(c) 解释回归直线中：(i) bb 的值；(ii) aa 的值。

(d) 估计总楼面面积为 4600 平方米的建筑物每年的供暖费用。

解答

(a)

解法一

思路

展开

因为 ∑y=0\sum y=0，所以 Sxy=∑xy−∑x∑ynS_{xy}=\sum xy-\frac{\sum x\sum y}{n} 会直接等于 ∑xy\sum xy。

答题过程

展开 Sxy=∑xy−∑x∑yn=1818−5(0)8=1818.S_{xy}=\sum xy-\frac{\sum x\sum y}{n} =1818-\frac{5(0)}{8}=1818.

(b)

解法一

思路

展开

先求 coded regression line y=c+bxy=c+bx，再把 x=f−350080x=\frac{f-3500}{80}、y=h−450080y=\frac{h-4500}{80} 代回去。

答题过程

展开

For the regression line of yy on xx,

b=SxySxx=18181754=1.03649….\begin{align*} b=\frac{S_{xy}}{S_{xx}}=\frac{1818}{1754}=1.03649\ldots. \end{align*}

Also,

xˉ=58,yˉ=0.\begin{align*} \bar x=\frac58,\qquad \bar y=0. \end{align*}

So

c=yˉ−bxˉ=0−(1.03649…)58=−0.6478….c=\bar y-b\bar x =0-(1.03649\ldots)\frac58 =-0.6478\ldots.

Thus

y=−0.6478…+1.03649…x.\begin{align*} y=-0.6478\ldots+1.03649\ldots x. \end{align*}

Substitute

y=h−450080,x=f−350080.\begin{align*} y=\frac{h-4500}{80},\qquad x=\frac{f-3500}{80}. \end{align*}

Then

h−450080=−0.6478…+1.03649…(f−350080).\frac{h-4500}{80} =-0.6478\ldots +1.03649\ldots\bigg(\frac{f-3500}{80}\bigg).

Multiplying by 8080,

h−4500=−51.82…+1.03649…(f−3500).\begin{align*} h-4500=-51.82\ldots+1.03649\ldots(f-3500). \end{align*}

Therefore

h=820.46…+1.03649…f.\begin{align*} h=820.46\ldots+1.03649\ldots f. \end{align*}

So the regression line is

h=820+1.04f.\begin{align*} h=820+1.04f. \end{align*}

(c)

(i)

思路

展开

斜率表示楼面面积每增加 1 m21\,\mathrm{m}^2，每年供暖费用的平均变化。

答题过程

展开

The value of bb means that for each extra 1 m21\,\mathrm{m}^2 of floor area, the annual heating bill increases by about \1.04$.

(ii)

思路

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截距表示模型在楼面面积为 0 时预测的基本供暖费用。

答题过程

展开

The value of aa means that the model predicts a base annual heating bill of about \820whenthefloorareaiswhen the floor area is0,\mathrm{m}^2$.

(d)

解法一

思路

展开

把 f=4600f=4600 代入回归线即可。

答题过程

展开

Using

h=820.46…+1.03649…f,\begin{align*} h=820.46\ldots+1.03649\ldots f, \end{align*}

when f=4600f=4600,

h=820.46…+1.03649…(4600)=5588.31….\begin{align*} h=820.46\ldots+1.03649\ldots(4600)=5588.31\ldots. \end{align*}

So the estimated annual heating bill is approximately

$5590.\begin{align*} \$5590. \end{align*}