Question

the electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature (x1) and the number of days in the month (x2). the past year’s historical data are available and are presented in the following table: y (electric power consumption) x1 (temperature) x2 (no of days) 240 25 24 236 31 21 270 45 24 274 60 25 301 65 25 316 72 26 300 80 25 296 84 25 267 75 24 276 60 25 288 50 25 261 38 23 i. find the equation of least squares regression line ŷ = a + bx1. ii. find the equation of least squares regression line ŷ = a + bx2. iii. for each regression line, interpret the values of a and b. iv. what measurement should be calculated to determine which variables (x1 or x2) that best describe y? calculate the measurement and interpret it. v. which variables (x1 or x2) that best describe y? why do you choose this variable?

Answer 1

The given data is:

y	x1	x2
240	25	24
236	31	21
270	45	24
274	60	25
301	65	25
316	72	26
300	80	25
296	84	25
267	75	24
276	60	25
288	50	25
261	38	23

i. find the equation of least squares regression line ŷ = a + bx1

First let us estimate the equation

For this, we need to have the the sum of squares and cross products.Let us create the following table for the same:

	x1	y	x1^2	y^2	x1*y
	25	240	625	57600	6000
	31	236	961	55696	7316
	45	270	2025	72900	12150
	60	274	3600	75076	16440
	65	301	4225	90601	19565
	72	316	5184	99856	22752
	80	300	6400	90000	24000
	84	296	7056	87616	24864
	75	267	5625	71289	20025
	60	276	3600	76176	16560
	50	288	2500	82944	14400
	38	261	1444	68121	9918
Total	685	3325	43245	927875	193990

We have n=12

The estimate

The estimate of a,

The correlation coefficient r between y and X1 is  

  

  

  

Coefficient of determination is

The equation of regression line is

ii. find the equation of least squares regression line ŷ = a + bx2.

	x2	y	x2^2	y^2	x2*y
	24	240	576	57600	5760
	21	236	441	55696	4956
	24	270	576	72900	6480
	25	274	625	75076	6850
	25	301	625	90601	7525
	26	316	676	99856	8216
	25	300	625	90000	7500
	25	296	625	87616	7400
	24	267	576	71289	6408
	25	276	625	76176	6900
	25	288	625	82944	7200
	23	261	529	68121	6003
Total	292	3325	7124	927875	81198

Coefficient of determination is

The equation of least squares regression line is

iii. for each regression line, interpret the values of a and b.

The equation with power and average ambient temperature is

Here, we have the intercept as 219.38. This is the value of the power when teperature is zero or in other words, this is the overhead charges. The slope is 1.0109 which is the increase in power for unit increase in temperature.

The equation relating power and the number of days is .

Here theoretically when number of days in a month is zero, the power consumption is -100.5167. For every additional day of work, the raise in power is 15.5178.

iv. what measurement should be calculated to determine which variables (x1 or x2) that best describe y? calculate the measurement and interpret it.

Here, we measure the coefficient of determination which describes how much of the variation in power consumption is explained by the liner models.

For the model involving X1, the . ie 64.41% of the variation in power consumption is explained by the average ambient temperature.

For the model involving X2, the . ie 68.39% of the variation in power consumption is explained by the number of days in the month.

v. which variables (x1 or x2) that best describe y? why do you choose this variable?

From the foregoing, we see that the for the model with X2 was more and hence, I choose the variable X2 since it best describe the model.