Chapter 4.6, Problem 1CYU

a.

To determine

To write: The equation of the best fit line.

To find: The correlation coefficient.

Answer to Problem 1CYU

The equation of best line is $y=1.179x+11$ .

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

Explanation of Solution

Given Information: A table showing the number of art students who use the pottery studio each day.

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.

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

$x$	$y$	$xy$	$x^2$
1	10	10	1
2	15	30	4
3	18	54	9
4	15	60	16
5	13	65	25
6	19	114	36
7	20	140	49

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=28\\ {\displaystyle \sum y}=110\\ {\displaystyle \sum x}y=473\\ {\displaystyle \sum x^2}=140\end{array}$

Calculate a as,

  $\begin{array}{c}a=\frac{{\displaystyle \sum y\cdot {{\displaystyle \sum x}}^2}-{\displaystyle \sum x\cdot {\displaystyle \sum x}y}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{110\times 140-28\times 473}{7\times 140-{28}^2}\\ =11\end{array}$

Calculate b as,

  $\begin{array}{c}b=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}\\ =\frac{7\times 473-28\times 110}{7\times 140-{28}^2}\\ =1.179\end{array}$

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

  $\begin{array}{c}y=ax+b\\ y=1.179x+11\end{array}$

Hence, the equation of best line is,

  $y=1.179x+11$

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

$x$	$y$	$xy$	$x^2$	$y^2$
1	10	10	1	100
2	15	30	4	225
3	18	54	9	324
4	15	60	16	225
5	13	65	25	169
6	19	114	36	361
7	20	140	49	400

Sum of every column is,

  $\begin{array}{c}{\displaystyle \sum x}=28\\ {\displaystyle \sum y}=110\\ {\displaystyle \sum x}y=473\\ {\displaystyle \sum x^2}=140\\ {\displaystyle \sum y^2}=1804\end{array}$

Calculate r as,

  $\begin{array}{c}r=\frac{n\cdot {\displaystyle \sum xy}-{\displaystyle \sum x\cdot {\displaystyle \sum y}}}{\sqrt{\left[n\cdot {\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2\right].\left[n\cdot {\displaystyle \sum y^2}-{\left({\displaystyle \sum y}\right)}^2\right]}}\\ =\frac{7\times 473-28\times 110}{\sqrt{\left[7\times 140-{28}^2\right]-\left[7\times 1804-{110}^2\right]}}\\ =0.7181\end{array}$

Therefore, the correlation coefficient is 0.7181

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

b.

To determine

To calculate: The regression line fits the model.

Answer to Problem 1CYU

The regression line does not explain the dataset very well.

Explanation of Solution

Given Information: A table showing the number of art students who use the pottery studio each day.

Calculation:

The graph is,

  

The line does not pass from any of the point. Though the line of best fit is moderately positively correlated, the regression line does not explain the dataset very well.