Question

The following information is given.

Details	X (in ₹)	Y (in ₹)
Arithmetic Mean	6	8
Standard Deviation	5	${\displaystyle \frac{40}{3}}$

Coefficient of correlation between X and Y is ${\displaystyle \frac{8}{15}}$. Find

1. The regression Coefficient of Y on X
2. The most likely value of Y when X = ₹ 100.

Answer 1

Given ${\displaystyle \overline{\text{X}}}$ = 6, ${\displaystyle \overline{\text{Y}}}$ = 8

σ_X = 5, σ_Y = ${\displaystyle \frac{40}{3}}$

and r(X, Y) = ${\displaystyle \frac{8}{15}}$

Regression coefficient of Y on X

b_yx = ${\displaystyle \text{r}.\frac{{\sigma }_{\text{X}}}{{\sigma }_{\text{Y}}}=\frac{8}{15}\left(\frac{40}{3\times 5}\right)}$

= ${\displaystyle \frac{320}{225}}$

= 1.422

b_yx = 1.422

∴ Regression line of Y on X is

${\displaystyle \text{Y}-\overline{\text{Y}}={\text{b}}_{\text{yx}}(\text{X}-\overline{\text{X}})}$

Y − 8 = 1.422 (X − 6)

Y − 8 = 1.422X − 8.532

Y = 1.422X − 0.532

When X = ₹ 100

Y = 1.422(100) − 0.532

= 142.2 − 0.532

Y = 141.67

∴ When X = ₹ 100, Y = ₹ 141.67