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IAL 2022 Oct Q2

A Level / Edexcel / S1

IAL 2022 Oct Paper · Question 2

题目

Problem

The production cost, £cc million, of a film and the total ticket sales, £tt million, earned by the film are recorded for a sample of 4040 films. Some summary statistics are given below.

∑c=1634∑t=1361∑t2=82873\sum c=1634\qquad \sum t=1361\qquad \sum t^2=82873 ∑ct=83634Scc=28732.1\sum ct=83634\qquad S_{cc}=28732.1

(a) Find the exact value of SttS_{tt} and the exact value of SctS_{ct}

(3)

(b) Calculate the value of the product moment correlation coefficient for these data.

(2)

(c) Give an interpretation of your answer to part (b)

(1)

(d) Show that the equation of the linear regression line of tt on cc can be written as

t=−5.84+0.976ct=-5.84+0.976c

where the values of the intercept and gradient are given to 33 significant figures.

(3)

(e) Find the expected total ticket sales for a film with a production cost of £9090 million.

(2)

Using the regression line in part (d)

(f) find the range of values of the production cost of a film for which the total ticket sales are less than 80%80\% of its production cost.

(2)

解答

(a)

解法一

思路

展开

直接用 SS 的定义式：平方和减去校正项。

答题过程

展开 Stt= ∑t2−(∑t)2n= 82873−1361240= 36564.975.\begin{align*} S_{tt} =&\,\sum t^2-\frac{(\sum t)^2}{n}\\[3mm] =&\,82873-\frac{1361^2}{40}\\[3mm] =&\,36564.975. \end{align*}

Also,

Sct= ∑ct−∑c∑tn= 83634−1634(1361)40= 28037.15.\begin{align*} S_{ct} =&\,\sum ct-\frac{\sum c\sum t}{n}\\[3mm] =&\,83634-\frac{1634(1361)}{40}\\[3mm] =&\,28037.15. \end{align*}

(b)

解法一

思路

展开

代入 PMCC 公式。

答题过程

展开 r= SctSccStt= 28037.1528732.1(36564.975)= 0.865…\begin{align*} r =&\,\frac{S_{ct}}{\sqrt{S_{cc}S_{tt}}}\\[3mm] =&\,\frac{28037.15}{\sqrt{28732.1(36564.975)}}\\[3mm] =&\,0.865\ldots \end{align*}

So

r=0.865\begin{align*} r=0.865 \end{align*}

to 33 significant figures.

(c)

解法一

思路

展开

rr 是正且较接近 11，说明成本越高，票房销售额一般越高。

答题过程

展开

There is a positive correlation: films with higher production costs generally have higher total ticket sales.

(d)

解法一

思路

展开

回归线 tt on cc 的斜率是 Sct/SccS_{ct}/S_{cc}，截距是 tˉ−bcˉ\bar{t}-b\bar{c}。

答题过程

展开 b= SctScc= 28037.1528732.1= 0.97581…\begin{align*} b =&\,\frac{S_{ct}}{S_{cc}}\\[3mm] =&\,\frac{28037.15}{28732.1}\\[3mm] =&\,0.97581\ldots \end{align*}

Also,

a= tˉ−bcˉ= 136140−0.97581…(163440)= −5.8369…\begin{align*} a =&\,\bar{t}-b\bar{c}\\[3mm] =&\,\frac{1361}{40}-0.97581\ldots\left(\frac{1634}{40}\right)\\[3mm] =&\,-5.8369\ldots \end{align*}

Therefore, to 33 significant figures,

t=−5.84+0.976c.\begin{align*} t=-5.84+0.976c. \end{align*}

(e)

解法一

思路

展开

把 c=90c=90 代入回归线。单位仍是 million pounds。

答题过程

展开 t= −5.84+0.976(90)= 82.0.\begin{align*} t =&\,-5.84+0.976(90)\\[3mm] =&\,82.0. \end{align*}

The expected total ticket sales are approximately

£82 million.\begin{align*} \text{£}82\text{ million}. \end{align*}

(f)

解法一

思路

展开

“ticket sales less than 80%80\% of production cost” 即 t<0.8ct<0.8c，再用回归线代入。

答题过程

展开

We need

t<0.8c.t<0.8c.

Using the regression line,

−5.84+0.976c<0.8c0.176c<5.84c<33.1818…\begin{align*} -5.84+0.976c&<0.8c\\[3mm] 0.176c&<5.84\\[3mm] c&<33.1818\ldots \end{align*}

Therefore

c<33.2.c<33.2.

So the production cost is less than about £33.233.2 million.