Question

Sales of Body	Price of Lens	Gender	Sales of Lens
155	$700	1	122
101	650	1	120
157	725	0	135
180	575	1	95
150	600	0	100
201	750	0	174
99	560	1	118
137	500	0	130
155	675	1	128
165	550	1	166
152	725	0	131
127	750	1	102
217	565	0	165
186	670	0	154
176	600	1	97
123	585	0	129
109	645	0	98
90	575	1	105
176	660	0	120
129	650	1	105

1. Find the regression coefficients and the regression equation for this data by using the method of least squares analysis.
2. Provide meanings for each of the regression coefficients. Be sure these meanings relate specifically to the context of the problem.
3. Use your regression equation to predict the sales of the lens when sales of the camera body are 200 and the price of the lens is $800. Do the same when the sales of the camera body are 150 and the price of the lens is $700. Do you have any concerns with either of these predictions? State a reason or reasons for any concerns you might have

Answer 1

Solution-:

Let, x1=Sales of Body , x2=Price of Lens and y=Sales of Lens

By using R-Software:

> x1=c(155,101,157,180,150,201,99,137,155,165,152,127,217,186,176,123,109,90,176,129);x1
[1] 155 101 157 180 150 201 99 137 155 165 152 127 217 186 176 123 109 90 176
[20] 129
> x2=c(700,650,725,575,600,750,560,500,675,550,725,750,565,670,600,585,645,575,660,650);x2
[1] 700 650 725 575 600 750 560 500 675 550 725 750 565 670 600 585 645 575 660
[20] 650
> y=c(122,120,135,95,100,174,118,130,128,166,131,102,165,154,97,129,98,105,120,105);y
[1] 122 120 135 95 100 174 118 130 128 166 131 102 165 154 97 129 98 105 120
[20] 105
> n=20
> #For (a)   
> fit=lm(y~x1+x2);fit

Call:
lm(formula = y ~ x1 + x2)

Coefficients:
(Intercept) x1 x2
66.394585 0.382100 0.002009

> #The estimated regression equation of y on x1 & x2 is: y= b0+b1*x1+b2*x2
> #The estimated regression equation of y on x1 & x2 is:y=66.3946+0.3821*x1+0.0020*X2
> #(b)Interpretation of coefficients:
> #b0, the y-intercept, can be interpreted as the value you would predict for y if both x1=0 and x2=0.
> #We would expect an average change of 66.3946 for Sales of Lens to Price of Lens and Sales of Body
> # We observed that in above equation unit change in Sales of Body will result into 0.3821 in value of dependent variable Sales of Lens.
> # And 0.0020 measures the change in the value of Sales of Lens for unit change in Price of Lens.
> #(c)
> #When x1=200, x2=800 and then, y=66.3946+0.3821*200+0.0020*800=144.41 approx =144
> #When x1=150, x2=700 and then, y=66.3946+0.3821*150+0.0020*700=125.11 approx =125

># We seen that Sales of Body and Price of Lens decreases then Sales of Lens is decreases.

R-Code:

x1=c(155,101,157,180,150,201,99,137,155,165,152,127,217,186,176,123,109,90,176,129);x1
x2=c(700,650,725,575,600,750,560,500,675,550,725,750,565,670,600,585,645,575,660,650);x2
y=c(122,120,135,95,100,174,118,130,128,166,131,102,165,154,97,129,98,105,120,105);y
n=20
#For (a)   
fit=lm(y~x1+x2);fit
#The estimated regression equation of y on x1 & x2 is: y= b0+b1*x1+b2*x2
#The estimated regression equation of y on x1 & x2 is:y=66.3946+0.3821*x1+0.0020*X2
#(b)Interpretation of coefficients:
#b0, the y-intercept, can be interpreted as the value you would predict for y if both x1=0 and x2=0.
#We would expect an average change of 66.3946 for Sales of Lens to Price of Lens and Sales of Body
# We observed that in above equation unit change in Sales of Body will result into 0.3821 in value of dependent variable Sales of Lens
# And 0.0020 measures the change in the value of Sales of Lens for unit change in Price of Lens
#(c)
#When x1=200, x2=800 and then, y=66.3946+0.3821*200+0.0020*800=144.41 approx =144
#When x1=150, x2=700 and then, y=66.3946+0.3821*150+0.0020*700=125.11 approx =125
# We seen that Sales of Body and Price of Lens decreases then Sales of Lens is decreases.