Question

the number of murders and robberies per 100,000 population for a random selection of states is show.

a.Find the correlation coefficient.

b.Find the equation of the regression line

c.Find the value for y when x=4.5 using the regression equation from above. Assume there is a relationship.

Murders 2.4 2.7 5.6 2.6 2.1 3.3 6.6 5.7

Robberies 25.3 14.3 151.6 91.1 80 49 173 95.8

Answer 1

X :- Murders

Y :- Robberies

ΣX = 31
ΣY = 680.1
ΣX * Y = 3202.71
ΣX2 = 142.52
ΣY2 = 80033.99

Correlation coefficient

((N + ΣΧΥ) - ΣΧΣΥ)/((ΝΣ(Χ2) - (ΣΧ)2] (ΝΣ(Y2) – (ΣΥ)2)

$r = ( ( 8 * 3202.71 ) - 31 * 680.1 ) / ( \sqrt{ ( 8 * 142.52 ) - ( 31 )^{2} )( 8 * 80033.99 ) - ( 680.1 )^{2} })$
r = 0.804

Equation of regression line is Ŷ = a + bX
$b = ( n \Sigma XY - \Sigma X \Sigma Y) / ( n\Sigma X^{2} - (\Sigma X)^{2})$

b = 25.333
a =( Σ Y - ( b * Σ X) ) / n
a =( 680.1 - ( 25.3326 * 31 ) ) / 8
a = -13.151
Equation of regression line becomes Ŷ = -13.151 + 25.333 X

When X = 4.5
Ŷ = -13.151 + 25.333 X
Ŷ = -13.151 + ( 25.333 * 4.5 )
Ŷ = 100.85