题目
Problem
The production cost, £c c c t t t 40 40 40
∑ c = 1634 ∑ t = 1361 ∑ t 2 = 82873 \sum c=1634\qquad
\sum t=1361\qquad
\sum t^2=82873 ∑ c = 1634 ∑ t = 1361 ∑ t 2 = 82873 ∑ c t = 83634 S c c = 28732.1 \sum ct=83634\qquad
S_{cc}=28732.1 ∑ c t = 83634 S cc = 28732.1 (a) Find the exact value of S t t S_{tt} S tt S c t S_{ct} S c t
(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 t t t c c c
t = − 5.84 + 0.976 c t=-5.84+0.976c t = − 5.84 + 0.976 c where the values of the intercept and gradient are given to 3 3 3
(3)
(e) Find the expected total ticket sales for a film with a production cost of £90 90 90
(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\% 80%
(2)
题目中文翻译
对一个由 40 40 40 c c c t t t
(a) 求 S t t S_{tt} S tt S c t S_{ct} S c t
(b) 计算这些数据的积矩相关系数。
(c) 解释你在 (b) 中所得答案的含义。
(d) 证明 t t t c c c
t = − 5.84 + 0.976 c , t=-5.84+0.976c, t = − 5.84 + 0.976 c , 其中截距和斜率均保留 3 3 3
(e) 求制作成本为 £90 90 90
使用 (d) 中的回归直线：
(f) 求电影制作成本的取值范围，使总票房小于制作成本的 80 % 80\% 80%
解答
(a)
解法一
思路
展开
直接用 S S S
答题过程
展开
S t t = ∑ t 2 − ( ∑ t ) 2 n = 82873 − 1361 2 40 = 36564.975. \begin{align*}
S_{tt}
=&\,\sum t^2-\frac{(\sum t)^2}{n}\\
=&\,82873-\frac{1361^2}{40}\\
=&\,36564.975.
\end{align*} S tt = = = ∑ t 2 − n ( ∑ t ) 2 82873 − 40 136 1 2 36564.975. Also,
S c t = ∑ c t − ∑ c ∑ t n = 83634 − 1634 ( 1361 ) 40 = 28037.15. \begin{align*}
S_{ct}
=&\,\sum ct-\frac{\sum c\sum t}{n}\\
=&\,83634-\frac{1634(1361)}{40}\\
=&\,28037.15.
\end{align*} S c t = = = ∑ c t − n ∑ c ∑ t 83634 − 40 1634 ( 1361 ) 28037.15.
(b)
解法一
思路
展开
代入 PMCC 公式。
答题过程
展开
r = S c t S c c S t t = 28037.15 28732.1 ( 36564.975 ) = 0.865 … \begin{align*}
r
=&\,\frac{S_{ct}}{\sqrt{S_{cc}S_{tt}}}\\
=&\,\frac{28037.15}{\sqrt{28732.1(36564.975)}}\\
=&\,0.865\ldots
\end{align*} r = = = S cc S tt S c t 28732.1 ( 36564.975 ) 28037.15 0.865 … So
r = 0.865 \begin{align*}
r=0.865
\end{align*} r = 0.865 to 3 3 3
(c)
解法一
思路
展开
r r r 1 1 1
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 t t t c c c S c t / S c c S_{ct}/S_{cc} S c t / S cc t ˉ − b c ˉ \bar{t}-b\bar{c} t ˉ − b c ˉ
答题过程
展开
b = S c t S c c = 28037.15 28732.1 = 0.97581 … \begin{align*}
b
=&\,\frac{S_{ct}}{S_{cc}}\\
=&\,\frac{28037.15}{28732.1}\\
=&\,0.97581\ldots
\end{align*} b = = = S cc S c t 28732.1 28037.15 0.97581 … Also,
a = t ˉ − b c ˉ = 1361 40 − 0.97581 … ( 1634 40 ) = − 5.8369 … \begin{align*}
a
=&\,\bar{t}-b\bar{c}\\
=&\,\frac{1361}{40}-0.97581\ldots\left(\frac{1634}{40}\right)\\
=&\,-5.8369\ldots
\end{align*} a = = = t ˉ − b c ˉ 40 1361 − 0.97581 … ( 40 1634 ) − 5.8369 … Therefore, to 3 3 3
t = − 5.84 + 0.976 c . \begin{align*}
t=-5.84+0.976c.
\end{align*} t = − 5.84 + 0.976 c .
(e)
解法一
思路
展开
把 c = 90 c=90 c = 90
答题过程
展开
t = − 5.84 + 0.976 ( 90 ) = 82.0. \begin{align*}
t
=&\,-5.84+0.976(90)\\
=&\,82.0.
\end{align*} t = = − 5.84 + 0.976 ( 90 ) 82.0. The expected total ticket sales are approximately
£ 82 million . \begin{align*}
\text{£}82\text{ million}.
\end{align*} £ 82 million .
(f)
解法一
思路
展开
“ticket sales less than 80 % 80\% 80% t < 0.8 c t<0.8c t < 0.8 c
答题过程
展开
We need
t < 0.8 c . t<0.8c. t < 0.8 c . Using the regression line,
− 5.84 + 0.976 c < 0.8 c 0.176 c < 5.84 c < 33.1818 … \begin{align*}
-5.84+0.976c&<0.8c\\
0.176c&<5.84\\
c&<33.1818\ldots
\end{align*} − 5.84 + 0.976 c 0.176 c c < 0.8 c < 5.84 < 33.1818 … Therefore
c < 33.2. c<33.2. c < 33.2. So the production cost is less than about £33.2 33.2 33.2
