Chapter 4.6, Problem 1BGP

To determine

To write: An equation of the best fit line

To find: The correlation coefficient

Answer to Problem 1BGP

The equation of best line is $y=5.952x+44.214$ .

The correlation coefficient is 0.8495. The relation is strongly positively correlated.

Explanation of Solution

Given Information: A table showing the number of goals of leading scorers for the Mustang Girls Hockey Team.

Calculation:

An equation of the best fit line is given by,

  $y=ax+b$ …… (1)

• Here, a is the intercept and y is the slope.

The years start from 2007 and go on till 2014.

Assume the year 2007 as 1 and x be the number of years since 2007.

Find $xy\text{ and }{x}^2$ ,

$x$	$y$	$xy$	$x^2$
1	63	63	1
2	44	88	4
3	55	165	9
4	63	252	16
5	81	405	25
6	85	510	36
7	93	651	49
8	84	672	64

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=36\\ {\displaystyle \sum y}=568\\ {\displaystyle \sum x}y=2806\\ {\displaystyle \sum x^2}=204\end{array}$

Calculate a as,

  $\begin{array}{c}a=\frac{{\displaystyle \sum y\cdot {{\displaystyle \sum x}}^2}-{\displaystyle \sum x\cdot {\displaystyle \sum x}y}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{568\times 204-36\times 2806}{8\times 204-{36}^2}\\ =44.214\end{array}$

Calculate b as,

  $\begin{array}{c}b=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{8\times 2806-36\times 568}{8\times 204-{36}^2}\\ =5.952\end{array}$

Substitute the values of a and b in equation (1),

  $\begin{array}{c}y=ax+b\\ y=5.952x+44.214\end{array}$

• Here, x be the number of years since 2007.

Hence, the equation of best line is,

  $y=5.952x+44.214$

The equation of best line is shown below:

  

Find $xy\text{ , }{x}^2\text{ and }{y}^2$ ,

$x$	$y$	$xy$	$x^2$	$y^2$
1	63	63	1	3960
2	44	88	4	1936
3	55	165	9	3025
4	63	252	16	3969
5	81	405	25	6561
6	85	510	36	7225
7	93	651	49	8649
8	84	672	64	7056

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=36\\ {\displaystyle \sum y}=281\\ {\displaystyle \sum x}y=1343\\ {\displaystyle \sum x^2}=204\\ {\displaystyle \sum y^2}=42390\end{array}$

Calculate r as,

  $\begin{array}{c}r=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{\sqrt{\left[n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right].\left[n\cdot {\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2\right]}}\\ =\frac{8\times 2806-36\times 568}{\sqrt{\left[8\times 204-{36}^2\right]-\left[8\times 42390-{568}^2\right]}}\\ =0.8495\end{array}$

Therefore, the correlation coefficient is 0.8495

The correlation coefficient lies between $-1\text{ and }1$ , here since the value is close to 1 the relation is strongly positively correlated.