Question

The data below relates diamond carats to purchase prices.

Carat	0.31	0.32	0.36	0.38	0.4	0.43	0.45	0.48	0.5
Price	1641	1468	1685	1420	1797	1824	2043	2342	2122

1. Run a linear regression model, with the carat being the independent variable ( X ) and the purchase price being the dependent variable ( Y ).

2. 1. (a) Find the estimate of the intercept for the linear regression model.

   2. (b) Find the estimate of the slope for the linear regression model.

   3. (c) What is the predicted price of a 0.37 carat diamond?

   4. (d) What fraction of the dependent variable’s variation can be explained by this linear regression model?

Answer 1

SOLUTION:

Given in the question
Carat is the independent variable and the purchase price is dependent variable
Regression equation can be calculated as
Y = a+bX
where a is intercept of regression line and b is slope of regression line
Slope of regression line can be calculated as
Slope = ((n*Summation(XY)) - (Summation(X)*Summation(Y))/(n*Summation(X^2) - (Summation(X))^2))

Carat(X)	Price(Y)	X^2	Y^2	XY
0.31	1641	0.0961	2692881	508.71
0.32	1468	0.1024	2155024	469.76
0.36	1685	0.1296	2839225	606.6
0.38	1420	0.1444	2016400	539.6
0.4	1797	0.16	3229209	718.8
0.43	1824	0.1849	3326976	784.32
0.45	2043	0.2025	4173849	919.35
0.48	2342	0.2304	5484964	1124.16
0.5	2122	0.25	4502884	1061
3.63	16342	1.5003	30421412	6732.3

Slope = (9*6732.3 - 3.63*16342) / (9*1.5003 - 3.63*3.63) = 3895.76
Intercept of regression line can be calculated as
Interecept = (Summation(Y) - slope*Summation(X))/n = (16342 - 3895.76*3.63)/9 = 244.49
So regression equation is Y = 244.49 + 3895.76*X
Solution(c)
If X = 0.37 than Y can be calcualted as

Y = 244.49 + 3895.76*X = 244.49 + 3895.76*0.37 = 1685.92
Solution(d)
For calculating coefficient of determination, first we will calculate correaltion coefficient which can be calculated as
Correlation coefficient = (n*Summation(XY) - Summation(X)*Summation(Y))/sqrt(((n*Summation(X^2) - Summation(X)^2))*((n*Summation(Y^2) - Summation(Y)^2))) = (9*6732.3 - 3.63*16342)/sqrt((9*1.5003 - 3.63*3.63)*(9*30421412 - 16342*16342)) = 1269.24/sqrt(0.3258*6731744) = 0.8570
So coefficient of determination can be calculated as
Coefficient of determination = (Correlation coeffcient)^2 = (0.8570)^2 = 0.7345
So this model explain the 73.45% fraction of the dependent variables variation can be explained by this linear regression model.