题目
Problem
A biologist is studying bears.
The biologist records the length, d d d g g g 8 8 8
∑ d = 1456.8 , ∑ g = 713.2 , ∑ d g = 141978.84 , ∑ g 2 = 72675.98 \sum d=1456.8,\quad \sum g=713.2,\quad
\sum dg=141978.84,\quad \sum g^2=72675.98 ∑ d = 1456.8 , ∑ g = 713.2 , ∑ d g = 141978.84 , ∑ g 2 = 72675.98 S d d = 16769.78. S_{dd}=16769.78. S dd = 16769.78. (a) Calculate the exact value of S d g S_{dg} S d g S g g S_{gg} S g g
(3)
(b) Calculate the value of the product moment correlation coefficient between d d d g g g
(2)
(c) Show that the equation of the regression line of g g g d d d
g = − 42.3 + 0.722 d g=-42.3+0.722d g = − 42.3 + 0.722 d where the values of the intercept and gradient are given to 3 3 3
(3)
(d) Give an interpretation, in context, of the gradient of the regression line.
(1)
Using the equation of the regression line given in part (c)
(e) (i) estimate the girth of a bear with a length of 2.5 2.5 2.5
(ii) explain why an estimate for the girth of a bear with a length of 0.5 0.5 0.5
(2)
Using the regression line from part (c), the biologist estimates that for each x x x 17.3 17.3 17.3
(f) Find the value of x x x
(2)
题目中文翻译
一名生物学家研究熊，并记录 8 8 8 d d d g g g
(a) 计算 S d g S_{dg} S d g S g g S_{gg} S g g
(b) 计算 d d d g g g
(c) 证明 g g g d d d g = − 42.3 + 0.722 d g=-42.3+0.722d g = − 42.3 + 0.722 d 3 3 3
(d) 结合情境解释回归直线斜率的意义。
利用 (c) 的回归直线：
(e) (i) 估计身长为 2.5 2.5 2.5
(ii) 解释为什么对身长为 0.5 0.5 0.5
利用 (c) 的回归直线，生物学家估计熊的身长每增加 x x x 17.3 17.3 17.3
(f) 求 x x x
解答
(a)
解法一
思路
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用
S d g = ∑ d g − ∑ d ∑ g n , \begin{align*}
S_{dg}=\sum dg-\frac{\sum d\sum g}{n},
\end{align*} S d g = ∑ d g − n ∑ d ∑ g , 以及
S g g = ∑ g 2 − ( ∑ g ) 2 n . \begin{align*}
S_{gg}=\sum g^2-\frac{(\sum g)^2}{n}.
\end{align*} S g g = ∑ g 2 − n ( ∑ g ) 2 .
答题过程
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S d g = 141978.84 − 1456.8 ( 713.2 ) 8 = 12105.12. \begin{align*}
S_{dg}
=&\,141978.84-\frac{1456.8(713.2)}{8}\\
=&\,12105.12.
\end{align*} S d g = = 141978.84 − 8 1456.8 ( 713.2 ) 12105.12. Also,
S g g = 72675.98 − 713.2 2 8 = 9094.2. \begin{align*}
S_{gg}
=&\,72675.98-\frac{713.2^2}{8}\\
=&\,9094.2.
\end{align*} S g g = = 72675.98 − 8 713. 2 2 9094.2.
(b)
解法一
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PMCC 的公式是
r = S d g S d d S g g . \begin{align*}
r=\frac{S_{dg}}{\sqrt{S_{dd}S_{gg}}}.
\end{align*} r = S dd S g g S d g .
答题过程
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r = S d g S d d S g g = 12105.12 16769.78 ( 9094.2 ) = 0.9802 … . \begin{align*}
r
=&\,\frac{S_{dg}}{\sqrt{S_{dd}S_{gg}}}\\
=&\,\frac{12105.12}{\sqrt{16769.78(9094.2)}}\\
=&\,0.9802\ldots.
\end{align*} r = = = S dd S g g S d g 16769.78 ( 9094.2 ) 12105.12 0.9802 … . Therefore
r = 0.980 \begin{align*}
r=0.980
\end{align*} r = 0.980 to 3 3 3
(c)
解法一
思路
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回归线 g = a + b d g=a+bd g = a + b d
b = S d g S d d . \begin{align*}
b=\frac{S_{dg}}{S_{dd}}.
\end{align*} b = S dd S d g . 再用均值点 ( d ˉ , g ˉ ) (\bar d,\bar g) ( d ˉ , g ˉ )
答题过程
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The gradient is
b = S d g S d d = 12105.12 16769.78 = 0.7218 … . \begin{align*}
b
=&\,\frac{S_{dg}}{S_{dd}}\\
=&\,\frac{12105.12}{16769.78}\\
=&\,0.7218\ldots.
\end{align*} b = = = S dd S d g 16769.78 12105.12 0.7218 … . Also,
d ˉ = 1456.8 8 = 182.1 , g ˉ = 713.2 8 = 89.15. \bar d=\frac{1456.8}{8}=182.1,\qquad
\bar g=\frac{713.2}{8}=89.15. d ˉ = 8 1456.8 = 182.1 , g ˉ = 8 713.2 = 89.15. So
a = g ˉ − b d ˉ = 89.15 − ( 0.7218 … ) ( 182.1 ) = − 42.297 … . \begin{align*}
a
=&\,\bar g-b\bar d\\
=&\,89.15-(0.7218\ldots)(182.1)\\
=&\,-42.297\ldots.
\end{align*} a = = = g ˉ − b d ˉ 89.15 − ( 0.7218 … ) ( 182.1 ) − 42.297 … . Therefore, to 3 3 3
g = − 42.3 + 0.722 d . \begin{align*}
g=-42.3+0.722d.
\end{align*} g = − 42.3 + 0.722 d .
(d)
解法一
思路
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斜率表示 d d d 1 1 1 g g g
答题过程
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For each 1 1 1 0.722 0.722 0.722
(e)(i)
解法一
思路
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注意单位：2.5 2.5 2.5 250 250 250
答题过程
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2.5 2.5 2.5 250 250 250
Using
g = − 42.3 + 0.722 d , \begin{align*}
g=-42.3+0.722d,
\end{align*} g = − 42.3 + 0.722 d , with d = 250 d=250 d = 250
g = − 42.3 + 0.722 ( 250 ) = 138.2. \begin{align*}
g
=&\,-42.3+0.722(250)\\
=&\,138.2.
\end{align*} g = = − 42.3 + 0.722 ( 250 ) 138.2. The estimated girth is
138 cm \begin{align*}
138\text{ cm}
\end{align*} 138 cm to 3 3 3
(e)(ii)
解法一
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0.5 0.5 0.5 50 50 50
答题过程
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0.5 0.5 0.5 50 50 50
Substituting d = 50 d=50 d = 50
g = − 42.3 + 0.722 ( 50 ) = − 6.2. \begin{align*}
g=-42.3+0.722(50)=-6.2.
\end{align*} g = − 42.3 + 0.722 ( 50 ) = − 6.2. This is a negative girth, which is not possible, so the estimate is not reliable.
(f)
解法一
思路
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斜率是每 1 1 1 x x x 0.722 x 0.722x 0.722 x
答题过程
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Using the gradient,
0.722 x = 17.3. \begin{align*}
0.722x=17.3.
\end{align*} 0.722 x = 17.3. Therefore
x = 17.3 0.722 = 23.96 … . \begin{align*}
x
=&\,\frac{17.3}{0.722}\\
=&\,23.96\ldots.
\end{align*} x = = 0.722 17.3 23.96 … . So
x = 24 \begin{align*}
x=24
\end{align*} x = 24 to 2 2 2
