Question

The following table shows the amount of converted sugar in a chemical process at different temperatures.

TEMPERATURE X CONVERTED SUGAR Y
1.2 8.2
1.4 8.5
1.6 8.4
1.8 9.3
2.0 8.9
2.2 10.5
2.4 9.3






a. Compute for the correlation coefficient.
b. Estimate the linear regression line.
c. Estimate the mean amount of sugar produced when the temperature recorded is 1.7.

Answer 1

a)x_i=1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4

$\mu_X = 12.6/7 = 1.8$

y_i=8.2, 8.5, 8.4, 9.3, 8.9, 10.5, 9.3

$\mu_Y = 63.1/7 = 9.0143$

$\sum(x_i - \mu_X)^2=1.12$

$\sum(y_i - \mu_Y)^2=3.68857$

$σ_X=\sqrt{\frac{1.12}{7}}=\sqrt{0.16}=0.4$

$σ_Y=\sqrt{\frac{3.68857}{7}}=\sqrt{0.5269}=0.7259$

$ρ_(XY)=\frac{1}{7}\times \frac{1.56}{0.4\times0.7259}=0.7675$

b) Sum of X = 12.6

Sum of Y = 63.1

Mean X = 1.8

Mean Y = 9.0143

Sum of squares (SS_X) = 1.12

Sum of products (SP) = 1.56

Regression Equation = ŷ = bX + a

b = SP/SS_X = 1.56/1.12 = 1.39286

a = M_Y - bM_X = 9.01 - (1.39*1.8) = 6.50714

ŷ = 1.39286X + 6.50714

c) using line formula, ŷ (1.7)=1.39286*1.7 + 6.50714=8.875