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

Solution

a. Correlation coefficient, r.

$$r={n \sum (xy)- (\sum x)( \sum y) \over \sqrt{[n\sum x^2 - ({\sum x})^2][n\sum y^2 - ({\sum y})^2]}}$$ $$={(7*115.14)-(12.6*63.1) \over \sqrt{[7*23.8 - {(12.6)}^2][7*572.49-{(63.1)}^2]}}$$ $$={10.92 \over 14.2277} = 0.76751$$

Ans: 0.76751

b. Linear regression line

$y=a+bx$

$$b= {n \sum (xy)- (\sum x)( \sum y) \over [n\sum x^2 - ({\sum x})^2]}$$ $$={7*115.14 - (12.6*63.1) \over (7*23.8 - {(12.6)}^2} = {10.92 \over 7.84 }$$ $$=1.3929$$

$$a= {\sum y - b\sum x \over n}$$ $$={63.1 - 1.3929(12.6) \over 7}$$ $$=6.5071$$

$\therefore y=6.5071+1.3929x$

c. mean amount of sugar when temp. is 1.7

$$y=6.5071+1.3929(1.7)$$ $$=8.87503$$