Chapter 1, Problem 69SGA

To determine

To find: the equation for regression line and also explain how representative line is of the data.

Answer to Problem 69SGA

  $y=-0.284x+12.964$

The correlation is slightly negative.

Explanation of Solution

Given information:

Table shows the data collected from ten students reveals the time spend on watching TV and on reading each week.

Watching TV	Reading
20	8.5
32	3.0
42	1.0
12	4.0
5	14.0
28	4.5
33	7.0
18	12.0
30	3.0
25	3.0

N =10

Calculation:

Create a table as follows:

$x$ (Watching TV)	$y$ (Reading)	$xy$	$x^2$
20	8.5	170	400
32	3.0	96	1024
42	1.0	42	1764
12	4.0	48	144
5	14.0	70	25
28	4.5	126	784
33	7.0	231	1089
18	12.0	216	324
30	3.0	90	900
25	3.0	75	625
${\displaystyle \sum x=}245$	${\displaystyle \sum y=}60$	${\displaystyle \sum xy=}1164$	${\displaystyle \sum x^2=}7079$

The slope of regression can be evaluated as follows:

  $m=\frac{\left(N{\displaystyle \sum xy}-{\displaystyle \sum x}\right)\left({\displaystyle \sum y}\right)}{N{\displaystyle \sum x^2}-{\left({\displaystyle \sum x}\right)}^2}$

Substitute the values and calculate the slope as follows:

  $\begin{array}{c}m=\frac{\left(10\left(1164\right)-\left(245\right)\right)\left(60\right)}{\left(10\right)\left(7079\right)-{\left(245\right)}^2}\\ =\frac{11640-14700}{70790-60025}\\ =\frac{-3060}{10765}\\ =-0.284\end{array}$

The intercept of regression can be evaluated as follows:

  $b=\frac{\left({\displaystyle \sum y}\right)-\left(m{\displaystyle \sum x}\right)}{N}$

Substitute the values and calculate the slope as follows:

  $\begin{array}{c}b=\frac{\left(60\right)-\left(-0.284\right)\left(245\right)}{10}\\ =\frac{60+68.58}{10}\\ =\frac{129.6}{10}\\ =12.96\end{array}$

The regression equation of the given data can be evaluated by substituting the values as shown below.

  $\begin{array}{l}y=mx+b\\ y=-0.284x+12.964\end{array}$

Thus, the regression equation is $y=-0.284x+12.964$ . The correlation is slightly negative.