Chapter 4.6, Problem 1AGP

To determine

To write: An equation of the best fit line

To find: The correlation coefficient

Answer to Problem 1AGP

The equation of best line is $y=1.869x+26.714$ .

The correlation coefficient is 0.6142. The relation is moderately 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	30	30	1
2	23	46	4
3	41	123	9
4	35	140	16
5	31	155	25
6	43	258	36
7	33	231	49
8	45	360	64

Sum of every column is,

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

Calculate a as,

  $\begin{array}{c}a=\frac{{\displaystyle \sum y\cdot {{\displaystyle \sum x}}^2}-{\displaystyle \sum x\cdot {\displaystyle \sum x}y}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{281\times 204-36\times 1343}{8\times 204-{36}^2}\\ =26.71\end{array}$

Calculate b as,

  $\begin{array}{c}b=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{8\times 1343-36\times 281}{8\times 204-{36}^2}\\ =1.869\end{array}$

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

  $\begin{array}{c}y=ax+b\\ y=1.869x+26.714\end{array}$

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

Hence, the equation of best line is,

  $y=1.869x+26.714$

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	30	30	1	900
2	23	46	4	529
3	41	123	9	1681
4	35	140	16	1225
5	31	155	25	961
6	43	258	36	1849
7	33	231	49	1089
8	45	360	64	2025

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=36\\ {\displaystyle \sum y}=281\\ {\displaystyle \sum x}y=1343\\ {\displaystyle \sum x^2}=204\\ {\displaystyle \sum y^2}=10259\\ \end{array}$

Calculate r as,

  $\begin{array}{c}r=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{\sqrt{\left[n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right].\left[n\cdot {\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2\right]}}\\ =\frac{8\times 1343-36\times 1343}{\sqrt{\left[8\times 204-{36}^2\right]-\left[8\times 10259-{281}^2\right]}}\\ =0.6142\end{array}$

Therefore, the correlation coefficient is 0.6142.

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