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, $x$ , and the amount spent on paper in a year, $p$ ( $hundreds), in each of$ 12$ randomly selected offices were recorded.

The results are summarised in the following statistics.

\sum x=93\qquad S_{xx}=148.25\qquad \sum p=273

\sum p^2=6602.72\qquad \sum xp=2347

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

(1)

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

(3)

(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 $p=\dfrac45a+\dfrac12bx$ , where $a$ and $b$ are the values found in part (c).

Using the data for the $93$ employees from the $12$ offices,

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

(3)

解答

(a)

解法一

思路

展开

用 $S_{xp}=\sum xp-\dfrac{\sum x\sum p}{n}$ 。

答题过程

展开

\begin{align*} S_{xp} =&\,2347-\frac{93(273)}{12}\\[3mm] =&\,2347-2115.75\\[3mm] =&\,231.25. \end{align*}

(b)

解法一

思路

展开

先求 $S_{pp}$ ，再代入相关系数公式。

答题过程

展开

\begin{align*} S_{pp} =&\,6602.72-\frac{273^2}{12}\\[3mm] =&\,391.97. \end{align*}

Therefore

\begin{align*} r =&\,\frac{231.25}{\sqrt{148.25(391.97)}}\\[3mm] =&\,0.9593\ldots \end{align*}

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

(c)

解法一

思路

展开

回归线 $p$ on $x$ 的斜率是 $S_{xp}/S_{xx}$ ，截距是 $\bar{p}-b\bar{x}$ 。

答题过程

展开

\begin{align*} b =&\,\frac{S_{xp}}{S_{xx}}\\[3mm] =&\,\frac{231.25}{148.25}\\[3mm] =&\,1.55986\ldots \end{align*}

Also,

\begin{align*} a =&\,\frac{273}{12}-1.55986\ldots\left(\frac{93}{12}\right)\\[3mm] =&\,10.66\ldots \end{align*}

Therefore

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

(d)

解法一

思路

展开

$p$ 的单位是 hundreds of dollars，所以斜率 $1.56$ 表示每多一名员工，纸张花费平均增加 $156$ dollars。

答题过程

展开

For each extra employee, the amount spent on paper increases by about $156$ dollars per year on average.

(e)

解法一

思路

展开

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

答题过程

展开

The original mean amount spent is

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

Using the director’s model,

\begin{align*} \text{new mean }p =&\,\frac45(10.66\ldots)+\frac12(1.55986\ldots)\left(\frac{93}{12}\right)\\[3mm] =&\,14.573\ldots \end{align*}

Therefore the percentage saving is

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

So the estimated saving is about

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

IAL 2021 Oct Q2

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

(d)

解法一

思路

答题过程

(e)

解法一

思路

答题过程