题目
Problem
Stuart is investigating the relationship between Gross Domestic Product (GDP) and the size of the population for a particular country.
He takes a random sample of 9 9 9 t t t g g g
The data are summarised as
n = 9 ∑ t = 7.87 ∑ g = 144.84 ∑ g 2 = 3624.41 n=9\qquad
\sum t=7.87\qquad
\sum g=144.84\qquad
\sum g^2=3624.41 n = 9 ∑ t = 7.87 ∑ g = 144.84 ∑ g 2 = 3624.41 S t t = 1.29 S t g = 40.25 S_{tt}=1.29\qquad S_{tg}=40.25 S tt = 1.29 S t g = 40.25 (a) Calculate the product moment correlation coefficient between t t t g g g
(3)
(b) Give an interpretation of your product moment correlation coefficient.
(1)
(c) Find the equation of the least squares regression line of g g g t t t g = a + b t g=a+bt g = a + b t
(4)
(d) Give an interpretation of the value of b b b
(1)
(e) (i) Use the regression line from part (c) to estimate the GDP, in billions of dollars, for a population of 7000000 7000000 7000000
(2)
(ii) Comment on the reliability of your answer in part (i). Give a reason, in context, for your answer.
(1)
Using the regression line from part (c), Stuart estimates that for a population increase of x x x 0.1 0.1 0.1
(f) Find the value of x x x
(2)
题目中文翻译
Stuart 正在研究某个国家的国内生产总值（GDP）与人口规模之间的关系。他随机抽取 9 9 9 t t t g g g
数据汇总如下：
n = 9 ∑ t = 7.87 ∑ g = 144.84 ∑ g 2 = 3624.41 n=9\qquad
\sum t=7.87\qquad
\sum g=144.84\qquad
\sum g^2=3624.41 n = 9 ∑ t = 7.87 ∑ g = 144.84 ∑ g 2 = 3624.41 S t t = 1.29 S t g = 40.25 S_{tt}=1.29\qquad S_{tg}=40.25 S tt = 1.29 S t g = 40.25 (a) 计算 t t t g g g
(b) 解释你的积矩相关系数。
(c) 求 g g g t t t g = a + b t g=a+bt g = a + b t
(d) 解释回归直线中 b b b
(e) (i) 使用 (c) 中的回归直线，估计人口为 7000000 7000000 7000000
(ii) 评论 (i) 中估计值的可靠性，并结合题目情境说明理由。
Stuart 使用 (c) 中的回归直线估计：人口每增加 x x x 0.1 0.1 0.1
(f) 求 x x x
解答
(a)
解法一
思路
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先求 S g g S_{gg} S g g
答题过程
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S g g = ∑ g 2 − ( ∑ g ) 2 n = 3624.41 − 144.84 2 9 = 1293.4516 … \begin{align*}
S_{gg}
=&\,\sum g^2-\frac{(\sum g)^2}{n}\\
=&\,3624.41-\frac{144.84^2}{9}\\
=&\,1293.4516\ldots
\end{align*} S g g = = = ∑ g 2 − n ( ∑ g ) 2 3624.41 − 9 144.8 4 2 1293.4516 … Therefore
r = S t g S t t S g g = 40.25 1.29 ( 1293.4516 … ) = 0.985 … \begin{align*}
r
=&\,\frac{S_{tg}}{\sqrt{S_{tt}S_{gg}}}\\
=&\,\frac{40.25}{\sqrt{1.29(1293.4516\ldots)}}\\
=&\,0.985\ldots
\end{align*} r = = = S tt S g g S t g 1.29 ( 1293.4516 … ) 40.25 0.985 … So
r = 0.985. \begin{align*}
r=0.985.
\end{align*} r = 0.985.
(b)
解法一
思路
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r r r 1 1 1
答题过程
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As the population increases, the GDP generally increases.
(c)
解法一
思路
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回归线 g g g t t t S t g / S t t S_{tg}/S_{tt} S t g / S tt g ˉ − b t ˉ \bar{g}-b\bar{t} g ˉ − b t ˉ
答题过程
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b = S t g S t t = 40.25 1.29 = 31.20155 … \begin{align*}
b
=&\,\frac{S_{tg}}{S_{tt}}\\
=&\,\frac{40.25}{1.29}\\
=&\,31.20155\ldots
\end{align*} b = = = S tt S t g 1.29 40.25 31.20155 … Also,
a = g ˉ − b t ˉ = 144.84 9 − 31.20155 … ( 7.87 9 ) = − 11.19068 … \begin{align*}
a
=&\,\bar{g}-b\bar{t}\\
=&\,\frac{144.84}{9}-31.20155\ldots\left(\frac{7.87}{9}\right)\\
=&\,-11.19068\ldots
\end{align*} a = = = g ˉ − b t ˉ 9 144.84 − 31.20155 … ( 9 7.87 ) − 11.19068 … Thus
g = − 11.2 + 31.2 t . \begin{align*}
g=-11.2+31.2t.
\end{align*} g = − 11.2 + 31.2 t .
(d)
解法一
思路
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t t t g g g 1 1 1
答题过程
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For each increase of 1 1 1 31.2 31.2 31.2
(e)(i)
解法一
思路
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7000000 7000000 7000000 7 7 7 t = 7 t=7 t = 7
答题过程
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g = − 11.2 + 31.2 ( 7 ) = 207.2. \begin{align*}
g
=&\,-11.2+31.2(7)\\
=&\,207.2.
\end{align*} g = = − 11.2 + 31.2 ( 7 ) 207.2. The estimated GDP is
207 billion dollars \begin{align*}
207\text{ billion dollars}
\end{align*} 207 billion dollars to 3 3 3
(e)(ii)
解法一
思路
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样本的平均 population 只有 7.87 9 ≈ 0.874 \dfrac{7.87}{9}\approx0.874 9 7.87 ≈ 0.874 7 7 7
答题过程
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The estimate is unreliable because a population of 7 7 7
(f)
解法一
思路
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GDP 的增加量等于斜率乘以 population 增加量。
答题过程
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0.1 = 31.2 x . \begin{align*}
0.1=31.2x.
\end{align*} 0.1 = 31.2 x . So
x = 0.1 31.2 = 0.003205 … \begin{align*}
x=\frac{0.1}{31.2}=0.003205\ldots
\end{align*} x = 31.2 0.1 = 0.003205 … Therefore
x = 0.00321 \begin{align*}
x=0.00321
\end{align*} x = 0.00321 million people.
解法二
思路
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在同一条回归线上选取两个 GDP 相差 0.1 0.1 0.1 x x x
答题过程
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Let the two population values be t t t t + x t+x t + x
g ( t + x ) − g ( t ) = [ − 11.2 + 31.2 ( t + x ) ] − ( − 11.2 + 31.2 t ) = 31.2 x . \begin{align*}
g(t+x)-g(t)
=&\,\big[-11.2+31.2(t+x)\big]\\
&\,-\big(-11.2+31.2t\big)\\
=&\,31.2x.
\end{align*} g ( t + x ) − g ( t ) = = [ − 11.2 + 31.2 ( t + x ) ] − ( − 11.2 + 31.2 t ) 31.2 x . The increase in GDP is 0.1 0.1 0.1
31.2 x = 0.1 , x = 0.003205 … \begin{align*}
31.2x=&\,0.1,\\
x=&\,0.003205\ldots
\end{align*} 31.2 x = x = 0.1 , 0.003205 … Hence
x = 0.00321 \begin{align*}
x=0.00321
\end{align*} x = 0.00321 million people.
