题目
Problem
Morgan is investigating the body length, b b b 8 8 8
x = b − 21 2 x=\frac{b-21}{2} x = 2 b − 21 The results for the coded data are summarised below
∑ x = − 1.2 ∑ x 2 = 5.1 \sum x=-1.2\qquad \sum x^2=5.1 ∑ x = − 1.2 ∑ x 2 = 5.1 (a) Find the mean of b b b
(3)
(b) Find the standard deviation of b b b
(3)
A 9 9 9 9 9 9 ∑ x = 0 \sum x=0 ∑ x = 0
(c) find
(i) the body length of the 9 9 9
(ii) the standard deviation of x x x 9 9 9
(4)
题目中文翻译
Morgan 正在研究松鼠的身体长度 b b b 8 8 8
x = b − 21 2 x=\frac{b-21}{2} x = 2 b − 21 对每只松鼠的数据进行编码。编码后数据的汇总结果为
∑ x = − 1.2 ∑ x 2 = 5.1. \sum x=-1.2\qquad \sum x^2=5.1. ∑ x = − 1.2 ∑ x 2 = 5.1. (a) 求 b b b
(b) 求 b b b
样本中又加入第 9 9 9 9 9 9 ∑ x = 0 \sum x=0 ∑ x = 0
(c) 求：(i) 第 9 9 9 9 9 9 x x x
解答
(a)
解法一
思路
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由 x = b − 21 2 x=\dfrac{b-21}{2} x = 2 b − 21 b = 2 x + 21 b=2x+21 b = 2 x + 21 b ˉ = 2 x ˉ + 21 \bar{b}=2\bar{x}+21 b ˉ = 2 x ˉ + 21
答题过程
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x ˉ = − 1.2 8 = − 0.15. \begin{align*}
\bar{x}=\frac{-1.2}{8}=-0.15.
\end{align*} x ˉ = 8 − 1.2 = − 0.15. Since
b = 2 x + 21 , \begin{align*}
b=2x+21,
\end{align*} b = 2 x + 21 , we have
b ˉ = 2 ( − 0.15 ) + 21 = 20.7. \begin{align*}
\bar{b}=2(-0.15)+21=20.7.
\end{align*} b ˉ = 2 ( − 0.15 ) + 21 = 20.7. The mean body length is
20.7 cm . \begin{align*}
20.7\text{ cm}.
\end{align*} 20.7 cm .
解法二
思路
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由 b = 2 x + 21 b=2x+21 b = 2 x + 21 8 8 8
答题过程
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∑ b = 2 ∑ x + 21 ( 8 ) = 2 ( − 1.2 ) + 168 = 165.6. \begin{align*}
\sum b
=&\,2\sum x+21(8)\\
=&\,2(-1.2)+168\\
=&\,165.6.
\end{align*} ∑ b = = = 2 ∑ x + 21 ( 8 ) 2 ( − 1.2 ) + 168 165.6. Hence
b ˉ = 165.6 8 = 20.7 cm . \bar b=\frac{165.6}{8}=20.7\text{ cm}. b ˉ = 8 165.6 = 20.7 cm .
(b)
解法一
思路
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先求 coded variable x x x b = 2 x + 21 b=2x+21 b = 2 x + 21 2 2 2
答题过程
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σ x 2 = 5.1 8 − ( − 1.2 8 ) 2 = 0.615. \begin{align*}
\sigma_x^2
=&\,\frac{5.1}{8}-\left(\frac{-1.2}{8}\right)^2\\
=&\,0.615.
\end{align*} σ x 2 = = 8 5.1 − ( 8 − 1.2 ) 2 0.615. So
σ x = 0.615 = 0.7842 … \begin{align*}
\sigma_x=\sqrt{0.615}=0.7842\ldots
\end{align*} σ x = 0.615 = 0.7842 … Since b = 2 x + 21 b=2x+21 b = 2 x + 21
σ b = 2 σ x = 2 ( 0.7842 … ) = 1.568 … \begin{align*}
\sigma_b=2\sigma_x=2(0.7842\ldots)=1.568\ldots
\end{align*} σ b = 2 σ x = 2 ( 0.7842 … ) = 1.568 … The standard deviation of b b b
1.57 cm \begin{align*}
1.57\text{ cm}
\end{align*} 1.57 cm to 3 3 3
解法二
思路
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由编码式展开 x 2 x^2 x 2 ∑ b 2 \sum b^2 ∑ b 2 b b b
答题过程
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Since b = 2 x + 21 b=2x+21 b = 2 x + 21 ∑ b = 165.6 \sum b=165.6 ∑ b = 165.6
5.1 = ∑ ( b − 21 2 ) 2 = 1 4 ( ∑ b 2 − 42 ∑ b + 8 ( 441 ) ) . \begin{align*}
5.1
=&\,\sum\left(\frac{b-21}{2}\right)^2\\
=&\,\frac14\left(\sum b^2-42\sum b+8(441)\right).
\end{align*} 5.1 = = ∑ ( 2 b − 21 ) 2 4 1 ( ∑ b 2 − 42 ∑ b + 8 ( 441 ) ) . Therefore
∑ b 2 = 3447.6. \sum b^2=3447.6. ∑ b 2 = 3447.6. Hence
σ b = 3447.6 8 − ( 165.6 8 ) 2 = 1.568 … = 1.57 cm \begin{align*}
\sigma_b
=&\,\sqrt{\frac{3447.6}{8}-\left(\frac{165.6}{8}\right)^2}\\
=&\,1.568\ldots\\
=&\,1.57\text{ cm}
\end{align*} σ b = = = 8 3447.6 − ( 8 165.6 ) 2 1.568 … 1.57 cm to 3 3 3
(c)(i)
解法一
思路
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原来 8 8 8 ∑ x = − 1.2 \sum x=-1.2 ∑ x = − 1.2 9 9 9 0 0 0 9 9 9 1.2 1.2 1.2
答题过程
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Let the coded value for the 9 9 9 x 9 x_9 x 9
− 1.2 + x 9 = 0 \begin{align*}
-1.2+x_9=0
\end{align*} − 1.2 + x 9 = 0 so
x 9 = 1.2. \begin{align*}
x_9=1.2.
\end{align*} x 9 = 1.2. Hence
b 9 = 2 ( 1.2 ) + 21 = 23.4. \begin{align*}
b_9=2(1.2)+21=23.4.
\end{align*} b 9 = 2 ( 1.2 ) + 21 = 23.4. The body length is
23.4 cm . \begin{align*}
23.4\text{ cm}.
\end{align*} 23.4 cm .
解法二
思路
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九个编码值的和为零，所以九只松鼠的平均编码值为零，对应的平均身体长度为 21 21 21
答题过程
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For all 9 9 9 x ˉ = 0 \bar x=0 x ˉ = 0 b ˉ = 21 \bar b=21 b ˉ = 21
∑ i = 1 9 b i = 9 ( 21 ) = 189. \sum_{i=1}^{9}b_i=9(21)=189. i = 1 ∑ 9 b i = 9 ( 21 ) = 189. The original eight body lengths have sum 8 ( 20.7 ) = 165.6 8(20.7)=165.6 8 ( 20.7 ) = 165.6
b 9 = 189 − 165.6 = 23.4 cm . b_9=189-165.6=23.4\text{ cm}. b 9 = 189 − 165.6 = 23.4 cm .
(c)(ii)
解法一
思路
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更新 ∑ x 2 \sum x^2 ∑ x 2 n = 9 n=9 n = 9 ∑ x = 0 \sum x=0 ∑ x = 0
答题过程
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For all 9 9 9
∑ x 2 = 5.1 + 1.2 2 = 6.54. \begin{align*}
\sum x^2=5.1+1.2^2=6.54.
\end{align*} ∑ x 2 = 5.1 + 1. 2 2 = 6.54. Also,
∑ x = 0. \begin{align*}
\sum x=0.
\end{align*} ∑ x = 0. Therefore
σ x = 6.54 9 − 0 2 = 0.852 … \begin{align*}
\sigma_x
=&\,\sqrt{\frac{6.54}{9}-0^2}\\
=&\,0.852\ldots
\end{align*} σ x = = 9 6.54 − 0 2 0.852 … So the standard deviation of x x x
0.852 \begin{align*}
0.852
\end{align*} 0.852 to 3 3 3
