Question

The population (in millions) and the violet crime rate (per 1000) were recorded for nine metropolitan areas . The data are shown in the following table.

Population (X)

0.4

10

1.3

2.1

7.0

0.3

0.3

0.2

0.2

Crime rate (Y)

6.9

12

9.5

9.2

8.4

7.3

7.1

7.0

6.9

a) obtain the estimated regression equation

b) make scatter diagram of the data above and graph the the predicted line

c) estimate the violet crime rate if the population of the area is 500,000.

d) compute the sample coefficient of determination and interpret the result.

Answer 1

a) Regression equation:

$y=a+bX$

Where;

a = intercept

b = slope coefficient

Following table shows the calculation of regression coefficients:

$a=\frac{[(\Sigma Y)(\Sigma X^{2})-(\Sigma X)(\Sigma XY)]}{[n(\Sigma X^{2})-(\Sigma X)^2]}$

$a=\frac{[(74.3)(155.5)-(21.8)(220.3)]}{[9(155.5)-(21.8)^2]}=7.3038$

$b=\frac{[n(\Sigma XY)-(\Sigma X)(\Sigma Y)]}{[n(\Sigma X^{2})-(\Sigma X)^2]}$

$b=\frac{[(9)(220.3)-(21.8)(74.3)]}{[9(155.5)-(21.8)^2]}=0.3929$

Regression equation is:

$y=7.3038+0.3929X$

b) Scatter diagram:

c) Estimation of violet crime rate:

 If the population of the area is 500,000 or 0.5 million then violet crime rate is:

$y=7.3038+0.3929*0.5$

$y=7.5 (per 1000)$

d) Coefficient of Determination:

$r^{2} =[ \frac{n(\Sigma XY)-(\Sigma X)(\Sigma Y)}{\sqrt{[n\Sigma X^{2}-(\Sigma X)^2]-[n\Sigma Y^{2}-(\Sigma Y)^2]}}]^{2}$

$r^{2}=[\frac{9(220.3)-(21.8)(74.3)}{\sqrt{[(9*155.5)-(21.8*21.8)]-[(9*637.4)-(74.3*74.3)]}}]^{2}$

$r^{2}=(0.813)^{2}=0.661$

The coefficient of determination is 0.661 that means 66.1% of the variation in the crime rate can be explained by the population. This indicates strong association between two variables.