Chapter 15
Multiple Regression
Learning Objectives
1 Understand how multiple regression analysis can be used to develop relationships involving one dependent variable and several independent variables
2 Be able to interpret the coefficients in a multiple regression analysis
3 Know the assumptions necessary to conduct statistical tests involving the hypothesized regression model
4 Understand the role of computer packages in performing multiple regression analysis
5 Be able to interpret and use computer output to develop the estimated regression equation
6 Be able to determine how good a fit is provided by the estimated regression equation
7 Be able to test for the significance of the regression equation
8 Understand how multicollinearity affects multiple regression analysis
9 Know how residual analysis can be used to make a judgement as to the appropriateness of the model identify outliers and determine which observations are influential
Solutions
1 a b1 5906 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 is held constant
b2 4980 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 is held constant
2 a The estimated regression equation is
4506 + 194x1
An estimate of y when x1 45 is
4506 + 194(45) 13236
b The estimated regression equation is
8522 + 432x2
An estimate of y when x2 15 is
8522 + 432(15) 15002
c The estimated regression equation is
1837 + 201x1 + 474x2
An estimate of y when x1 45 and x2 15 is
1837 + 201(45) + 474(15) 14318
3 a b1 38 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 x3 and x4
are held constant
b2 23 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 x3 and x4 are held constant
b3 76 is an estimate of the change in y corresponding to a 1 unit change in x3 when x1 x2 and x4 are held constant
b4 27 is an estimate of the change in y corresponding to a 1 unit change in x4 when x1 x2 and x3 are held constant
4 a 235 + 10(15) + 8(10) 255 sales estimate 255000
b Sales can be expected to increase by 10 for every dollar increase in inventory investment when advertising expenditure is held constant Sales can be expected to increase by 8 for every dollar increase in advertising expenditure when inventory investment is held constant
5 a The Minitab output is shown below
The regression equation is
Revenue 886 + 160 TVAdv
Predictor Coef SE Coef T P
Constant 88638 1582 5602 0000
TVAdv 16039 04778 336 0015
S 1215 RSq 653 RSq(adj) 595
Analysis of Variance
Source DF SS MS F P
Regression 1 16640 16640 1127 0015
Residual Error 6 8860 1477
Total 7 25500
b The Minitab output is shown below
The regression equation is
Revenue 832 + 229 TVAdv + 130 NewsAdv
Predictor Coef SE Coef T P
Constant 83230 1574 5288 0000
TVAdv 22902 03041 753 0001
NewsAdv 13010 03207 406 0010
S 06426 RSq 919 RSq(adj) 887
Analysis of Variance
Source DF SS MS F P
Regression 2 23435 11718 2838 0002
Residual Error 5 2065 0413
Total 7 25500
Source DF Seq SS
TVAdv 1 16640
NewsAdv 1 6795
c No it is 160 in part 2(a) and 299 above In this exercise it represents the marginal change in revenue due to an increase in television advertising with newspaper advertising held constant
d Revenue 832 + 229(35) + 130(18) 9356 or 93560
6 a The Minitab output is shown below
The regression equation is
Speed 498 + 00151 Weight
Predictor Coef SE Coef T P
Constant 4978 1911 261 0021
Weight 0015104 0006005 252 0025
S 7000 RSq 311 RSq(adj) 262
Analysis of Variance
Source DF SS MS F P
Regression 1 30995 30995 633 0025
Error 14 68600 4900
Total 15 99595
b The Minitab output is shown below
The regression equation is
Speed 805 000312 Weight + 0105 Horsepwr
Predictor Coef SE Coef T P
Constant 80487 9139 881 0000
Weight 0003122 0003481 090 0386
Horsepwr 010471 001331 786 0000
S 3027 RSq 880 RSq(adj) 862
Analysis of Variance
Source DF SS MS F P
Regression 2 87680 43840 4783 0000
Residual Error 13 11915 917
Total 15 99595
7 a The Minitab output is shown below
The regression equation is
Sales 665 + 0414 Compet 0270 Heller
Predictor Coef SE Coef T P
Constant 6652 4188 159 0156
Compet 04139 02604 159 0156
Heller 026978 008091 333 0013
S 1874 RSq 653 RSq(adj) 554
Analysis of Variance
Source DF SS MS F P
Regression 2 46188 23094 658 0025
Residual Error 7 24573 3510
Total 9 70761
b b1 414 is an estimate of the change in the quantity sold (1000s) of the Heller mower with respect to a 1 change in price in competitor’s mower with the price of the Heller mower held constant b2 270 is an estimate of the change in the quantity sold (1000s) of the Heller mower with respect to a 1 change in its price with the price of the competitor’s mower held constant
c 665 + 0414(170) 0270(160) 9368 or 93680 units
8 a The Minitab output is shown below
The regression equation is
Return 247 328 Safety + 346 ExpRatio
Predictor Coef SE Coef T P
Constant 2474 1104 224 0039
Safety 3284 1395 235 0031
ExpRatio 3459 1413 245 0026
S 1698 RSq 582 RSq(adj) 533
Analysis of Variance
Source DF SS MS F P
Regression 2 68232 34116 1184 0001
Residual Error 17 48997 2882
Total 19 117230
b
9 a The Minitab output is shown below
The regression equation is
College 267 143 Size + 00757 SatScore
Predictor Coef SE Coef T P
Constant 2671 5167 052 0613
Size 14298 09931 144 0170
SatScore 007574 003906 194 0072
S 1242 RSq 382 RSq(adj) 300
Analysis of Variance
Source DF SS MS F P
Regression 2 14304 7152 464 0027
Residual Error 15 23127 1542
Total 17 37431
b 267 143(20) + 00757(1000) 738
Estimate is 738
10 a The Minitab output is shown below
The regression equation is
Revenue 333 + 798 Cars
Predictor Coef SE Coef T P
Constant 3334 8308 040 0695
Cars 79840 06323 1263 0000
S 2267 RSq 925 RSq(adj) 919
Analysis of Variance
Source DF SS MS F P
Regression 1 8192067 8192067 15944 0000
Error 13 667936 51380
Total 14 8860003
b An increase of 1000 cars in service will result in an increase in revenue of 798 million
c The Minitab output is shown below
The regression equation is
Revenue 106 + 894 Cars 0191 Location
Predictor Coef SE Coef T P
Constant 10597 8552 124 0239
Cars 89427 07746 1155 0000
Location 01914 01026 187 0087
S 2077 RSq 942 RSq(adj) 932
Analysis of Variance
Source DF SS MS F P
Regression 2 8342186 4171093 9666 0000
Error 12 517817 43151
Total 14 8860003
11 a SSE SST SSR 6724125 6216375 50775
b
c
d The estimated regression equation provided an excellent fit
12 a
b
c Yes after adjusting for the number of independent variables in the model we see that 905 of the variability in y has been accounted for
13 a
b
c The estimated regression equation provided an excellent fit
14 a
b
c The adjusted coefficient of determination shows that 68 of the variability has been explained by the two independent variables thus we conclude that the model does not explain a large amount of variability
15 a
b Multiple regression analysis is preferred since both R2 andshow an increased percentage of the variability of y explained when both independent variables are used
16 Note the Minitab output is shown with the solution to Exercise 6
a No RSq 311
b Multiple regression analysis is preferred since both RSq and RSq(adj) show an increased percentage of the variability of y explained when both independent variables are used
17 a
b The fit is not very good
18 Note The Minitab output is shown with the solution to Exercise 10
a RSq 942 RSq(adj) 932
b The fit is very good
19 a MSR SSRp 62163752 3108188
b F MSRMSE 310818872536 4285
F05 474 (2 degrees of freedom numerator and 7 denominator)
Since F 4285 > F05 474 the overall model is significant
c t 59060813 726
t025 2365 (7 degrees of freedom)
Since t 2365 > t025 2365 b1 is significant
d t 49800567 878
Since t 878 > t025 2365 b2 is significant
20 A portion of the Minitab output is shown below
The regression equation is
Y 184 + 201 X1 + 474 X2
Predictor Coef SE Coef T P
Constant 1837 1797 102 0341
X1 20102 02471 813 0000
X2 47378 09484 500 0002
S 1271 RSq 926 RSq(adj) 904
Analysis of Variance
Source DF SS MS F P
Regression 2 140522 70261 4350 0000
Residual Error 7 11307 1615
Total 9 151829
a Since the pvalue corresponding to F 4350 is 000 < a 05 we reject H0 b1 b2 0 there is a significant relationship
b Since the pvalue corresponding to t 813 is 000 < a 05 we reject H0 b1 0 b1 is significant
c Since the pvalue corresponding to t 500 is 002 < a 05 we reject H0 b2 0 b2 is significant
21 a In the two independent variable case the coefficient of x1 represents the expected change in y corresponding to a one unit increase in x1 when x2 is held constant In the single independent variable case the coefficient of x1 represents the expected change in y corresponding to a one unit increase in x1
b Yes If x1 and x2 are correlated one would expect a change in x1 to be accompanied by a change in x2
22 a SSE SST SSR 16000 12000 4000
b F MSRMSE 600057143 1050
F05 474 (2 degrees of freedom numerator and 7 denominator)
Since F 1050 > F05 474 we reject H0 There is a significant relationship among the variables
23 a F 2838
F01 1327 (2 degrees of freedom numerator and 1 denominator)
Since F > F01 1327 reject H0
Alternatively the pvalue of 002 leads to the same conclusion
b t 753
t025 2571
Since t > t025 2571 b1 is significant and x1 should not be dropped from the model
c t 406
t025 2571
Since t > t025 2571 b2 is significant and x2 should not be dropped from the model
24 Note The Minitab output is shown in part (b) of Exercise 6
a F 4783
F05 381 (2 degrees of freedom numerator and 13 denominator)
Since F 4783 > F05 381 we reject H0 b1 b2 0
Alternatively since the pvalue 000 < a 05 we can reject H0
b For Weight
H0 b1 0 Ha b1 ¹ 0
Since the pvalue 0386 > a 005 we cannot reject H0
For Horsepower
H0 b2 0 Ha b2 ¹ 0
Since the pvalue 0000 < a 005 we can reject H0
25 a The Minitab output is shown below
The regression equation is
PE 604 + 0692 Profit + 0265 Sales
Predictor Coef SE Coef T P
Constant 6038 4589 132 0211
Profit 06916 02133 324 0006
Sales 02648 01871 142 0180
S 5456 RSq 472 RSq(adj) 390
Analysis of Variance
Source DF SS MS F P
Regression 2 34528 17264 580 0016
Residual Error 13 38700 2977
Total 15 73228
b Since the pvalue 0016 < a 005 there is a significant relationship among the variables
c For Profit Since the pvalue 0006 < a 005 Profit is significant
For Sales Since the pvalue 0180 > a 005 Sales is not significant
26 Note The Minitab output is shown with the solution to Exercise 10
a Since the pvalue corresponding to F 9666 is 0000 < a 05 there is a significant relationship among the variables
b For Cars Since the pvalue 0000 < a 005 Cars is significant
c For Location Since the pvalue 0087 > a 005 Location is not significant
27 a 291270 + 5906(180) + 4980(310) 2898150
b The point estimate for an individual value is 2898150 the same as the point estimate of the mean value
28 a Using Minitab the 95 confidence interval is 13216 to 15416
b Using Minitab the 95 prediction interval is 11113 to 17518
29 a 832 + 229(35) + 130(18) 93555 or 93555
Note In Exercise 5b the Minitab output also shows that b0 83230 b1 22902
and b2 13010 hence 83230 + 22902x1 + 13010x2 Using this estimated regression equation we obtain
83230 + 22902(35) + 13010(18) 93588 or 93588
The difference (93588 93555 33) is simply due to the fact that additional significant digits are used in the computations From a practical point of view however the difference is not enough to be concerned about In practice a computer software package is always used to perform the computations and this will not be an issue
The Minitab output is shown below
Fit StdevFit 95 CI 95 PI
93588 0291 ( 92840 94335) ( 91774 95401)
Note that the value of FIT () is 93588
b Confidence interval estimate 92840 to 94335 or 92840 to 94335
c Prediction interval estimate 91774 to 95401 or 91774 to 95401
30 a Since weight is not statistically significant (see Exercise 24) we will use an estimated regression equation which uses only Horsepower to predict the speed at 14 mile The Minitab output is shown below
The regression equation is
Speed 726 + 00968 Horsepwr
Predictor Coef SE Coef T P
Constant 72650 2655 2736 0000
Horsepwr 0096756 0009865 981 0000
S 3006 RSq 873 RSq(adj) 864
Analysis of Variance
Source DF SS MS F P
Regression 1 86943 86943 9621 0000
Residual Error 14 12652 904
Total 15 99595
Unusual Observations
Obs Horsepwr Speed Fit SE Fit Residual St Resid
2 290 108000 100709 0814 7291 252R
6 450 116200 116190 2036 0010 000 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence
The output shows that the point estimate is a speed of 101290 miles per hour
b The 95 confidence interval is 99490 to 103089 miles per hour
c The 95 prediction interval is 94596 to 107984 miles per hour
31 a Using Minitab the 95 confidence interval is 5837 to 7503
b Using Minitab the 95 prediction interval is 3524 to 9059
32 a E(y) b0 + b1 x1 + b2 x2 where
x2 0 if level 1 and 1 if level 2
b E(y) b0 + b1 x1 + b2(0) b0 + b1 x1
c E(y) b0 + b1 x1 + b2(1) b0 + b1 x1 + b2
d b2 E(y | level 2) E(y | level 1)
b1 is the change in E(y) for a 1 unit change in x1 holding x2 constant
33 a two
b E(y) b0 + b1 x1 + b2 x2 + b3 x3 where
x2
x3
Level
0
0
1
1
0
2
0
1
3
c E(y | level 1) b0 + b1 x1 + b2(0) + b3(0) b0+ b1 x1
E(y | level 2) b0 + b1 x1 + b2(1) + b3(0) b0 + b1 x1 + b2
E(y | level 3) b0 + b1 x1 + b2(0) + b3(0) b0 + b1 x1 + b3
b2 E(y | level 2) E(y | level 1)
b3 E(y | level 3) E(y | level 1)
b1 is the change in E(y) for a 1 unit change in x1 holding x2 and x3 constant
34 a 15300
b Estimate of sales 101 42(2) + 68(8) + 153(0) 561 or 56100
c Estimate of sales 101 42(1) + 68(3) + 153(1) 416 or 41600
35 a Let Type 0 if a mechanical repair
Type 1 if an electrical repair
The Minitab output is shown below
The regression equation is
Time 345 + 0617 Type
Predictor Coef SE Coef T P
Constant 34500 05467 631 0000
Type 06167 07058 087 0408
S 1093 RSq 87 RSq(adj) 00
Analysis of Variance
Source DF SS MS F P
Regression 1 0913 0913 076 0408
Residual Error 8 9563 1195
Total 9 10476
b The estimated regression equation did not provide a good fit In fact the pvalue of 408 shows that the relationship is not significant for any reasonable value of a
c Person 0 if Bob Jones performed the service and Person 1 if Dave Newton performed the service The Minitab output is shown below
The regression equation is
Time 462 160 Person
Predictor Coef SE Coef T P
Constant 46200 03192 1447 0000
Person 16000 04514 354 0008
S 07138 RSq 611 RSq(adj) 562
Analysis of Variance
Source DF SS MS F P
Regression 1 64000 64000 1256 0008
Residual Error 8 40760 05095
Total 9 104760
d We see that 611 of the variability in repair time has been explained by the repair person that performed the service an acceptable but not good fit
36 a The Minitab output is shown below
The regression equation is
Time 186 + 0291 Months + 110 Type 0609 Person
Predictor Coef SE Coef T P
Constant 18602 07286 255 0043
Months 029144 008360 349 0013
Type 11024 03033 363 0011
Person 06091 03879 157 0167
S 04174 RSq 900 RSq(adj) 850
Analysis of Variance
Source DF SS MS F P
Regression 3 94305 31435 1804 0002
Residual Error 6 10455 01743
Total 9 104760
b Since the pvalue corresponding to F 1804 is 002 < a 05 the overall model is statistically significant
c The pvalue corresponding to t 157 is 167 > a 05 thus the addition of Person is not statistically significant Person is highly correlated with Months (the sample correlation coefficient is 691) thus once the effect of Months has been accounted for Person will not add much to the model
37 a Let Position 0 if a guard
Position 1 if an offensive tackle
b The Minitab output is shown below
The regression equation is
Rating 112 + 0732 Position + 00222 Weight 228 Speed
Predictor Coef SE Coef T P
Constant 11223 4523 248 0022
Position 07324 02893 253 0019
Weight 002219 001039 214 0045
Speed 22775 09290 245 0023
S 06936 RSq 475 RSq(adj) 401
Analysis of Variance
Source DF SS MS F P
Regression 3 91562 30521 635 0003
Residual Error 21 101014 04810
Total 24 192576
c Since the pvalue corresponding to F 635 is 003 < a 05 there is a significant relationship between rating and the independent variables
d The value of RSq (adj) is 401 the estimated regression equation did not provide a very good fit
e Since the pvalue for Position is t 253 < a 05 position is a significant factor in the player’s rating
f
38 a The Minitab output is shown below
The regression equation is
Risk 918 + 108 Age + 0252 Pressure + 874 Smoker
Predictor Coef SE Coef T P
Constant 9176 1522 603 0000
Age 10767 01660 649 0000
Pressure 025181 004523 557 0000
Smoker 8740 3001 291 0010
S 5757 RSq 873 RSq(adj) 850
Analysis of Variance
Source DF SS MS F P
Regression 3 36607 12202 3682 0000
Residual Error 16 5302 331
Total 19 41909
b Since the pvalue corresponding to t 291 is 010 < a 05 smoking is a significant factor
c Using Minitab the point estimate is 3427 the 95 prediction interval is 2135 to 4718 Thus the probability of a stroke (2135 to 4718 at the 95 confidence level) appears to be quite high The physician would probably recommend that Art quit smoking and begin some type of treatment designed to reduce his blood pressure
39 a The Minitab output is shown below
The regression equation is
Y 020 + 260 X
Predictor Coef SE Coef T P
Constant 0200 2132 009 0931
X 26000 06429 404 0027
S 2033 RSq 845 RSq(adj) 793
Analysis of Variance
Source DF SS MS F P
Regression 1 67600 67600 1635 0027
Residual Error 3 12400 4133
Total 4 80000
b Using Minitab we obtained the following values
xi
yi
Standardized Residual
1
3
28
16
2
7
54
94
3
5
80
165
4
11
106
24
5
14
132
62
The point (35) does not appear to follow the trend of remaining data however the value of the standardized residual for this point 165 is not large enough for us to conclude that (3 5) is an outlier
c Using Minitab we obtained the following values
xi
yi
Studentized
Deleted Residual
1
3
13
2
7
91
3
5
442
4
11
19
5
14
54
t025 4303 (n p 2 5 1 2 2 degrees of freedom)
Since the studentized deleted residual for (3 5) is 442 < 4303 we conclude that the 3rd observation is an outlier
40 a The Minitab output is shown below
The regression equation is
Y 533 + 311 X
Predicator
Coef
Stdev
tratio
p
Constant
53280
5786
921
0003
X
31100
02016
1543
0001
s 2851 Rsq 988 Rsq (adj) 983
Analysis of Variance
SOURCE
DF
SS
MS
F
p
Regression
1
19344
19344
23803
0001
Error
3
244
81
Total
4
15988
b Using the Minitab we obtained the following values
xi
yi
Studentized
Deleted Residual
22
12
194
24
21
12
26
31
179
28
35
40
40
70
190
t025 4303 (n p 2 5 1 2 2 degrees of freedom)
Since none of the studentized deleted residuals are less than 4303 or greater than 4303 none of the observations can be classified as an outlier
c Using Minitab we obtained the following values
xi
yi
hi
22
12
38
24
21
28
26
31
22
28
35
20
40
70
92
The critical value is
Since none of the values exceed 12 we conclude that there are no influential observations in the data
d Using Minitab we obtained the following values
xi
yi
Di
22
12
60
24
21
00
26
31
26
28
35
03
40
70
1109
Since D5 1109 > 1 (rule of thumb critical value) we conclude that the fifth observation is influential
41 a The Minitab output appears in the solution to part (b) of Exercise 5 the estimated regression equation is
Revenue 832 + 229 TVAdv + 130 NewsAdv
b Using Minitab we obtained the following values
Standardized Residual
9663
162
9041
108
9434
122
9221
37
9439
110
9424
40
9442
112
9335
108
With the relatively few observations it is difficult to determine if any of the assumptions regarding the error term have been violated For instance an argument could be made that there does not appear to be any pattern in the plot alternatively an argument could be made that there is a curvilinear pattern in the plot
c The values of the standardized residuals are greater than 2 and less than +2 thus using test there are no outliers As a further check for outliers we used Minitab to compute the following studentized deleted residuals
Observation
Studentized Deleted Residual
1
211
2
110
3
131
4
33
5
113
6
36
7
116
8
110
t025 2776 (n p 2 8 2 2 4 degrees of freedom)
Since none of the studentized deleted residuals is less tan 2776 or greater than 2776 we conclude that there are no outliers in the data
d Using Minitab we obtained the following values
Observation
hi
Di
1
63
152
2
65
70
3
30
22
4
23
01
5
26
14
6
14
01
7
66
81
8
13
06
The critical average value is
Since none of the values exceed 1125 we conclude that there are no influential observations
However using Cook’s distance measure we see that D1 > 1 (rule of thumb critical value) thus we conclude the first observation is influential Final Conclusion observations 1 is an influential observation
42 a The Minitab output is shown below
The regression equation is
Speed 713 + 0107 Price + 00845 Horsepwr
Predictor Coef SE Coef T P
Constant 71328 2248 3173 0000
Price 010719 003918 274 0017
Horsepwr 0084496 0009306 908 0000
S 2485 RSq 919 RSq(adj) 907
Analysis of Variance
Source DF SS MS F P
Regression 2 91566 45783 7412 0000
Residual Error 13 8030 618
Total 15 99595
Source DF Seq SS
Price 1 40639
Horsepwr 1 50927
Unusual Observations
Obs Price Speed Fit SE Fit Residual St Resid
2 938 108000 105882 2007 2118 145 X
X denotes an observation whose X value gives it large influence
b The standardized residual plot is shown below There appears to be a very unusual trend in the standardized residuals
xx x
12+
x
SRES1 x
x
x
00+ x x
x
x x x
12+ x
x
x
+++++FITS1
900 960 1020 1080 1140
c The Minitab output shown in part (a) did not identify any observations with a large standardized residual thus there does not appear to be any outliers in the data
d The Minitab output shown in part (a) identifies observation 2 as an influential observation
43 a The Minitab output is shown below
The regression equation is
College 266 + 00970 SatScore
Predictor Coef SE Coef T P
Constant 2661 3722 072 0485
SatScore 009703 003734 260 0019
S 1283 RSq 297 RSq(adj) 253
Analysis of Variance
Source DF SS MS F P
Regression 1 11108 11108 675 0019
Residual Error 16 26323 1645
Total 17 37431
Unusual Observations
Obs SatScore College Fit SE Fit Residual St Resid
3 716 4000 4286 1079 286 041 X
X denotes an observation whose X value gives it large influence
b The Minitab output shown in part a identifies observation 3 as an influential observation
c The Minitab output appears in the solution to Exercise 9 the estimates regression equation is College 267 143 Size + 00757 SATScore
d The following Minitab output was also provided as part of the regression output for part c
Unusual Observations
Obs Size College Fit StdevFit Residual StResid
3 300 400 3804 1097 196 034 X
X denotes an obs whose X value gives it large influence
Observation 3 is still identified as an influential observation
44 a The expected increase in final college grade point average corresponding to a one point increase in high school grade point average is 0235 when SAT mathematics score does not change Similarly the expected increase in final college grade point average corresponding to a one point increase in the SAT mathematics score is 00486 when the high school grade point average does not change
b 141 + 0235(84) + 00486(540) 319
45 a Job satisfaction can be expected to decrease by 869 units with a one unit increase in length of service if the wage rate does not change A dollar increase in the wage rate is associated with a 135 point increase in the job satisfaction score when the length of service does not change
b 144 869(4) + 135(65) 6739
46 a The computer output with the missing values filled in is as follows
The regression equation is
Y 8103 + 7602 X1 + 3111 X2
Predicator
Coef
Stdev
tratio
Constant
8103
2667
304
X1
7602
2105
361
X2
3111
0613
508
s 335 Rsq 923 Rsq (adj) 910
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
1612
806
7182
Error
12
13467
112225
Total
14
174667
b t025 2179 (12 DF)
for b1 361 > 2179 reject H0 b1 0
for b2 508 > 2179 reject H0 b2 0
c See computer output
d
47 a The regression equation is
Y 141 + 0235 X1 + 00486 X2
Predictor
Coef
Stdev
tratio
Constant
14053
04848
290
X1
0023467
0008666
271
X2
00486
0001077
451
s 01298 Rsq 937 Rsq (adj) 919
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
176209
881
5244
Error
7
1179
0168
Total
9
188000
b F05 474 (2 DF numerator 7 DF denominator)
F 5244 > F05 significant relationship
c
good fit
d t025 2365 (7 DF)
for B1 t 271 > 2365 reject H0 B1 0
for B2 t 451 > 2365 reject H0 B2 0
48 a The regression equation is
Y 144 869 X1 + 1352 X2
Predictor
Coef
Stdev
tratio
Constant
14448
8191
176
X1
869
1555
559
X2
13517
2085
648
s 3773 Rsq 901 Rsq (adj) 861
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
64883
324415
2279
Error
5
7117
14234
Total
7
72000
b F05 579 (5 DF)
F 2279 > F05 significant relationship
c
good fit
d t025 2571 (5 DF)
for b1 t 559 < 2571 reject H0 b1 0
for b2 t 648 > 2571 reject H0 b2 0
49 a The Minitab output is shown below
The regression equation is
Price 128 + 226 BookVal
Predictor Coef SE Coef T P
Constant 12793 6624 193 0064
BookVal 22649 06631 342 0002
S 1950 RSq 294 RSq(adj) 269
Analysis of Variance
Source DF SS MS F P
Regression 1 44339 44339 1167 0002
Error 28 106423 3801
Total 29 150761
b The value of Rsq is 294 the estimated regression equation does not provide a good fit
c The Minitab output is shown below
The regression equation is
Price 588 + 254 BookVal + 0484 ReturnEq
Predictor Coef SE Coef T P
Constant 5877 5545 106 0299
BookVal 25356 05331 476 0000
ReturnEq 04841 01174 412 0000
S 1555 RSq 567 RSq(adj) 535
Analysis of Variance
Source DF SS MS F P
Regression 2 85442 42721 1766 0000
Error 27 65319 2419
Total 29 150761
Since the pvalue corresponding to the F test is 0000 the relationship is significant
50 a The Minitab output is shown below
The regression equation is
Speed 976 + 00693 Price 000082 Weight + 00590
Horsepwr 248 Zero60
Predictor Coef SE Coef T P
Constant 9757 1179 827 0000
Price 006928 003805 182 0096
Weight 0000816 0002593 031 0759
Horsepwr 005901 001543 382 0003
Zero60 24836 09601 259 0025
S 2127 RSq 950 RSq(adj) 932
Analysis of Variance
Source DF SS MS F P
Regression 4 94618 23655 5228 0000
Residual Error 11 4977 452
Total 15 99595
b Since the pvalue corresponding to the F test is 0000 the relationship is significant
c Since the pvalues corresponding to the t test for both Horsepwr (pvalue 003) and Zero60 (pvalue 025) are less than 05 both of these independent variables are significant
d The Minitab output is shown below
The regression equation is
Speed 103 + 00558 Horsepwr 319 Zero60
Predictor Coef SE Coef T P
Constant 103103 9448 1091 0000
Horsepwr 005582 001452 384 0002
Zero60 31876 09658 330 0006
S 2301 RSq 931 RSq(adj) 920
Analysis of Variance
Source DF SS MS F P
Regression 2 92712 46356 8754 0000
Residual Error 13 6884 530
Total 15 99595
Source DF Seq SS
Horsepwr 1 86943
Zero60 1 5768
Unusual Observations
Obs Horsepwr Speed Fit SE Fit Residual St Resid
2 290 108000 103352 1015 4648 225R
12 155 84600 82747 1773 1853 126 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence
e The standardized residual plot is shown below
SRES x
15+
x x
2 x x
00+ x x 2
x
xx
15+
x x
++++++FIT
840 900 960 1020 1080 1140
There is an unusual trend in the plot and one observation appears to be an outlier
f The Minitab output indicates that observation 2 is an outlier
g The Minitab output indicates that observation 12 is an influential observation
51 a The Minitab output is shown below
640+
x
Exposure
480+
x
x
320+
x
160+ x 3 x
x
++++++TimesAir
15 30 45 60 75 90
b The Minitab output is shown below
The regression equation is
Exposure 532 + 674 TimesAir
Predictor Coef SE Coef T P
Constant 5324 1653 322 0012
TimesAir 67427 04472 1508 0000
S 3170 RSq 966 RSq(adj) 962
Analysis of Variance
Source DF SS MS F P
Regression 1 228520 228520 22736 0000
Error 8 8041 1005
Total 9 236561
Since the pvalue is 0000 the relationship is significant
c The Minitab output is shown below
The regression equation is
Exposure 731 + 504 TimesAir + 101 BigAds
Predictor Coef SE Coef T P
Constant 73063 7507 973 0000
TimesAir 50368 03268 1541 0000
BigAds 10111 1599 632 0000
S 1308 RSq 995 RSq(adj) 993
Analysis of Variance
Source DF SS MS F P
Regression 2 235363 117682 68784 0000
Error 7 1198 171
Total 9 236561
d The pvalue corresponding to the t test for BigAds is 0000 thus the dummy variable is significant
e The dummy variable enables us to fit two different lines to the data this approach is referred to as piecewise linear approximation
52 a The Minitab output is shown below
Resale 388 +0000766 Price
Predictor Coef SE Coef T P
Constant 38772 4348 892 0000
Price 00007656 00001900 403 0000
S 5421 RSq 367 RSq(adj) 344
Analysis of Variance
Source DF SS MS F P
Regression 1 47725 47725 1624 0000
Residual Error 28 82292 2939
Total 29 130017
Since the pvalue corresponding to F 1624 is 000 < a 05 there is a significant relationship between Resale and Price
b RSq 367 not a very good fit
c Let Type1 0 and Type2 0 if a small pickup Type1 1 and Type2 0 if a fullsize pickup and Type1 0 and Type2 1 if a sport utility
The Minitab output using Type1 Type2 and Price is shown below
The regression equation is
Resale 426 + 909 Type1 + 792 Type2 +0000341 Price
Predictor Coef SE Coef T P
Constant 42554 3562 1195 0000
Type1 9090 2248 404 0000
Type2 7917 2163 366 0001
Price 00003415 00001800 190 0069
S 4298 RSq 631 RSq(adj) 588
Analysis of Variance
Source DF SS MS F P
Regression 3 81977 27326 1479 0000
Residual Error 26 48040 1848
Total 29 130017
d Since the pvalue corresponding to F 1479 is 000 < a 05 there is a significant relationship between Resale and the independent variables Note that individually Price is not significant at the 05 level of significance If we rerun the regression using just Type1 and Type2 the value of RSq (adj) decreases to 544 a drop of only 4 Thus it appears that for these data the type of vehicle is the strongest predictor of the resale value
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Multiple Regression
Learning Objectives
1 Understand how multiple regression analysis can be used to develop relationships involving one dependent variable and several independent variables
2 Be able to interpret the coefficients in a multiple regression analysis
3 Know the assumptions necessary to conduct statistical tests involving the hypothesized regression model
4 Understand the role of computer packages in performing multiple regression analysis
5 Be able to interpret and use computer output to develop the estimated regression equation
6 Be able to determine how good a fit is provided by the estimated regression equation
7 Be able to test for the significance of the regression equation
8 Understand how multicollinearity affects multiple regression analysis
9 Know how residual analysis can be used to make a judgement as to the appropriateness of the model identify outliers and determine which observations are influential
Solutions
1 a b1 5906 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 is held constant
b2 4980 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 is held constant
2 a The estimated regression equation is
4506 + 194x1
An estimate of y when x1 45 is
4506 + 194(45) 13236
b The estimated regression equation is
8522 + 432x2
An estimate of y when x2 15 is
8522 + 432(15) 15002
c The estimated regression equation is
1837 + 201x1 + 474x2
An estimate of y when x1 45 and x2 15 is
1837 + 201(45) + 474(15) 14318
3 a b1 38 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 x3 and x4
are held constant
b2 23 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 x3 and x4 are held constant
b3 76 is an estimate of the change in y corresponding to a 1 unit change in x3 when x1 x2 and x4 are held constant
b4 27 is an estimate of the change in y corresponding to a 1 unit change in x4 when x1 x2 and x3 are held constant
4 a 235 + 10(15) + 8(10) 255 sales estimate 255000
b Sales can be expected to increase by 10 for every dollar increase in inventory investment when advertising expenditure is held constant Sales can be expected to increase by 8 for every dollar increase in advertising expenditure when inventory investment is held constant
5 a The Minitab output is shown below
The regression equation is
Revenue 886 + 160 TVAdv
Predictor Coef SE Coef T P
Constant 88638 1582 5602 0000
TVAdv 16039 04778 336 0015
S 1215 RSq 653 RSq(adj) 595
Analysis of Variance
Source DF SS MS F P
Regression 1 16640 16640 1127 0015
Residual Error 6 8860 1477
Total 7 25500
b The Minitab output is shown below
The regression equation is
Revenue 832 + 229 TVAdv + 130 NewsAdv
Predictor Coef SE Coef T P
Constant 83230 1574 5288 0000
TVAdv 22902 03041 753 0001
NewsAdv 13010 03207 406 0010
S 06426 RSq 919 RSq(adj) 887
Analysis of Variance
Source DF SS MS F P
Regression 2 23435 11718 2838 0002
Residual Error 5 2065 0413
Total 7 25500
Source DF Seq SS
TVAdv 1 16640
NewsAdv 1 6795
c No it is 160 in part 2(a) and 299 above In this exercise it represents the marginal change in revenue due to an increase in television advertising with newspaper advertising held constant
d Revenue 832 + 229(35) + 130(18) 9356 or 93560
6 a The Minitab output is shown below
The regression equation is
Speed 498 + 00151 Weight
Predictor Coef SE Coef T P
Constant 4978 1911 261 0021
Weight 0015104 0006005 252 0025
S 7000 RSq 311 RSq(adj) 262
Analysis of Variance
Source DF SS MS F P
Regression 1 30995 30995 633 0025
Error 14 68600 4900
Total 15 99595
b The Minitab output is shown below
The regression equation is
Speed 805 000312 Weight + 0105 Horsepwr
Predictor Coef SE Coef T P
Constant 80487 9139 881 0000
Weight 0003122 0003481 090 0386
Horsepwr 010471 001331 786 0000
S 3027 RSq 880 RSq(adj) 862
Analysis of Variance
Source DF SS MS F P
Regression 2 87680 43840 4783 0000
Residual Error 13 11915 917
Total 15 99595
7 a The Minitab output is shown below
The regression equation is
Sales 665 + 0414 Compet 0270 Heller
Predictor Coef SE Coef T P
Constant 6652 4188 159 0156
Compet 04139 02604 159 0156
Heller 026978 008091 333 0013
S 1874 RSq 653 RSq(adj) 554
Analysis of Variance
Source DF SS MS F P
Regression 2 46188 23094 658 0025
Residual Error 7 24573 3510
Total 9 70761
b b1 414 is an estimate of the change in the quantity sold (1000s) of the Heller mower with respect to a 1 change in price in competitor’s mower with the price of the Heller mower held constant b2 270 is an estimate of the change in the quantity sold (1000s) of the Heller mower with respect to a 1 change in its price with the price of the competitor’s mower held constant
c 665 + 0414(170) 0270(160) 9368 or 93680 units
8 a The Minitab output is shown below
The regression equation is
Return 247 328 Safety + 346 ExpRatio
Predictor Coef SE Coef T P
Constant 2474 1104 224 0039
Safety 3284 1395 235 0031
ExpRatio 3459 1413 245 0026
S 1698 RSq 582 RSq(adj) 533
Analysis of Variance
Source DF SS MS F P
Regression 2 68232 34116 1184 0001
Residual Error 17 48997 2882
Total 19 117230
b
9 a The Minitab output is shown below
The regression equation is
College 267 143 Size + 00757 SatScore
Predictor Coef SE Coef T P
Constant 2671 5167 052 0613
Size 14298 09931 144 0170
SatScore 007574 003906 194 0072
S 1242 RSq 382 RSq(adj) 300
Analysis of Variance
Source DF SS MS F P
Regression 2 14304 7152 464 0027
Residual Error 15 23127 1542
Total 17 37431
b 267 143(20) + 00757(1000) 738
Estimate is 738
10 a The Minitab output is shown below
The regression equation is
Revenue 333 + 798 Cars
Predictor Coef SE Coef T P
Constant 3334 8308 040 0695
Cars 79840 06323 1263 0000
S 2267 RSq 925 RSq(adj) 919
Analysis of Variance
Source DF SS MS F P
Regression 1 8192067 8192067 15944 0000
Error 13 667936 51380
Total 14 8860003
b An increase of 1000 cars in service will result in an increase in revenue of 798 million
c The Minitab output is shown below
The regression equation is
Revenue 106 + 894 Cars 0191 Location
Predictor Coef SE Coef T P
Constant 10597 8552 124 0239
Cars 89427 07746 1155 0000
Location 01914 01026 187 0087
S 2077 RSq 942 RSq(adj) 932
Analysis of Variance
Source DF SS MS F P
Regression 2 8342186 4171093 9666 0000
Error 12 517817 43151
Total 14 8860003
11 a SSE SST SSR 6724125 6216375 50775
b
c
d The estimated regression equation provided an excellent fit
12 a
b
c Yes after adjusting for the number of independent variables in the model we see that 905 of the variability in y has been accounted for
13 a
b
c The estimated regression equation provided an excellent fit
14 a
b
c The adjusted coefficient of determination shows that 68 of the variability has been explained by the two independent variables thus we conclude that the model does not explain a large amount of variability
15 a
b Multiple regression analysis is preferred since both R2 andshow an increased percentage of the variability of y explained when both independent variables are used
16 Note the Minitab output is shown with the solution to Exercise 6
a No RSq 311
b Multiple regression analysis is preferred since both RSq and RSq(adj) show an increased percentage of the variability of y explained when both independent variables are used
17 a
b The fit is not very good
18 Note The Minitab output is shown with the solution to Exercise 10
a RSq 942 RSq(adj) 932
b The fit is very good
19 a MSR SSRp 62163752 3108188
b F MSRMSE 310818872536 4285
F05 474 (2 degrees of freedom numerator and 7 denominator)
Since F 4285 > F05 474 the overall model is significant
c t 59060813 726
t025 2365 (7 degrees of freedom)
Since t 2365 > t025 2365 b1 is significant
d t 49800567 878
Since t 878 > t025 2365 b2 is significant
20 A portion of the Minitab output is shown below
The regression equation is
Y 184 + 201 X1 + 474 X2
Predictor Coef SE Coef T P
Constant 1837 1797 102 0341
X1 20102 02471 813 0000
X2 47378 09484 500 0002
S 1271 RSq 926 RSq(adj) 904
Analysis of Variance
Source DF SS MS F P
Regression 2 140522 70261 4350 0000
Residual Error 7 11307 1615
Total 9 151829
a Since the pvalue corresponding to F 4350 is 000 < a 05 we reject H0 b1 b2 0 there is a significant relationship
b Since the pvalue corresponding to t 813 is 000 < a 05 we reject H0 b1 0 b1 is significant
c Since the pvalue corresponding to t 500 is 002 < a 05 we reject H0 b2 0 b2 is significant
21 a In the two independent variable case the coefficient of x1 represents the expected change in y corresponding to a one unit increase in x1 when x2 is held constant In the single independent variable case the coefficient of x1 represents the expected change in y corresponding to a one unit increase in x1
b Yes If x1 and x2 are correlated one would expect a change in x1 to be accompanied by a change in x2
22 a SSE SST SSR 16000 12000 4000
b F MSRMSE 600057143 1050
F05 474 (2 degrees of freedom numerator and 7 denominator)
Since F 1050 > F05 474 we reject H0 There is a significant relationship among the variables
23 a F 2838
F01 1327 (2 degrees of freedom numerator and 1 denominator)
Since F > F01 1327 reject H0
Alternatively the pvalue of 002 leads to the same conclusion
b t 753
t025 2571
Since t > t025 2571 b1 is significant and x1 should not be dropped from the model
c t 406
t025 2571
Since t > t025 2571 b2 is significant and x2 should not be dropped from the model
24 Note The Minitab output is shown in part (b) of Exercise 6
a F 4783
F05 381 (2 degrees of freedom numerator and 13 denominator)
Since F 4783 > F05 381 we reject H0 b1 b2 0
Alternatively since the pvalue 000 < a 05 we can reject H0
b For Weight
H0 b1 0 Ha b1 ¹ 0
Since the pvalue 0386 > a 005 we cannot reject H0
For Horsepower
H0 b2 0 Ha b2 ¹ 0
Since the pvalue 0000 < a 005 we can reject H0
25 a The Minitab output is shown below
The regression equation is
PE 604 + 0692 Profit + 0265 Sales
Predictor Coef SE Coef T P
Constant 6038 4589 132 0211
Profit 06916 02133 324 0006
Sales 02648 01871 142 0180
S 5456 RSq 472 RSq(adj) 390
Analysis of Variance
Source DF SS MS F P
Regression 2 34528 17264 580 0016
Residual Error 13 38700 2977
Total 15 73228
b Since the pvalue 0016 < a 005 there is a significant relationship among the variables
c For Profit Since the pvalue 0006 < a 005 Profit is significant
For Sales Since the pvalue 0180 > a 005 Sales is not significant
26 Note The Minitab output is shown with the solution to Exercise 10
a Since the pvalue corresponding to F 9666 is 0000 < a 05 there is a significant relationship among the variables
b For Cars Since the pvalue 0000 < a 005 Cars is significant
c For Location Since the pvalue 0087 > a 005 Location is not significant
27 a 291270 + 5906(180) + 4980(310) 2898150
b The point estimate for an individual value is 2898150 the same as the point estimate of the mean value
28 a Using Minitab the 95 confidence interval is 13216 to 15416
b Using Minitab the 95 prediction interval is 11113 to 17518
29 a 832 + 229(35) + 130(18) 93555 or 93555
Note In Exercise 5b the Minitab output also shows that b0 83230 b1 22902
and b2 13010 hence 83230 + 22902x1 + 13010x2 Using this estimated regression equation we obtain
83230 + 22902(35) + 13010(18) 93588 or 93588
The difference (93588 93555 33) is simply due to the fact that additional significant digits are used in the computations From a practical point of view however the difference is not enough to be concerned about In practice a computer software package is always used to perform the computations and this will not be an issue
The Minitab output is shown below
Fit StdevFit 95 CI 95 PI
93588 0291 ( 92840 94335) ( 91774 95401)
Note that the value of FIT () is 93588
b Confidence interval estimate 92840 to 94335 or 92840 to 94335
c Prediction interval estimate 91774 to 95401 or 91774 to 95401
30 a Since weight is not statistically significant (see Exercise 24) we will use an estimated regression equation which uses only Horsepower to predict the speed at 14 mile The Minitab output is shown below
The regression equation is
Speed 726 + 00968 Horsepwr
Predictor Coef SE Coef T P
Constant 72650 2655 2736 0000
Horsepwr 0096756 0009865 981 0000
S 3006 RSq 873 RSq(adj) 864
Analysis of Variance
Source DF SS MS F P
Regression 1 86943 86943 9621 0000
Residual Error 14 12652 904
Total 15 99595
Unusual Observations
Obs Horsepwr Speed Fit SE Fit Residual St Resid
2 290 108000 100709 0814 7291 252R
6 450 116200 116190 2036 0010 000 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence
The output shows that the point estimate is a speed of 101290 miles per hour
b The 95 confidence interval is 99490 to 103089 miles per hour
c The 95 prediction interval is 94596 to 107984 miles per hour
31 a Using Minitab the 95 confidence interval is 5837 to 7503
b Using Minitab the 95 prediction interval is 3524 to 9059
32 a E(y) b0 + b1 x1 + b2 x2 where
x2 0 if level 1 and 1 if level 2
b E(y) b0 + b1 x1 + b2(0) b0 + b1 x1
c E(y) b0 + b1 x1 + b2(1) b0 + b1 x1 + b2
d b2 E(y | level 2) E(y | level 1)
b1 is the change in E(y) for a 1 unit change in x1 holding x2 constant
33 a two
b E(y) b0 + b1 x1 + b2 x2 + b3 x3 where
x2
x3
Level
0
0
1
1
0
2
0
1
3
c E(y | level 1) b0 + b1 x1 + b2(0) + b3(0) b0+ b1 x1
E(y | level 2) b0 + b1 x1 + b2(1) + b3(0) b0 + b1 x1 + b2
E(y | level 3) b0 + b1 x1 + b2(0) + b3(0) b0 + b1 x1 + b3
b2 E(y | level 2) E(y | level 1)
b3 E(y | level 3) E(y | level 1)
b1 is the change in E(y) for a 1 unit change in x1 holding x2 and x3 constant
34 a 15300
b Estimate of sales 101 42(2) + 68(8) + 153(0) 561 or 56100
c Estimate of sales 101 42(1) + 68(3) + 153(1) 416 or 41600
35 a Let Type 0 if a mechanical repair
Type 1 if an electrical repair
The Minitab output is shown below
The regression equation is
Time 345 + 0617 Type
Predictor Coef SE Coef T P
Constant 34500 05467 631 0000
Type 06167 07058 087 0408
S 1093 RSq 87 RSq(adj) 00
Analysis of Variance
Source DF SS MS F P
Regression 1 0913 0913 076 0408
Residual Error 8 9563 1195
Total 9 10476
b The estimated regression equation did not provide a good fit In fact the pvalue of 408 shows that the relationship is not significant for any reasonable value of a
c Person 0 if Bob Jones performed the service and Person 1 if Dave Newton performed the service The Minitab output is shown below
The regression equation is
Time 462 160 Person
Predictor Coef SE Coef T P
Constant 46200 03192 1447 0000
Person 16000 04514 354 0008
S 07138 RSq 611 RSq(adj) 562
Analysis of Variance
Source DF SS MS F P
Regression 1 64000 64000 1256 0008
Residual Error 8 40760 05095
Total 9 104760
d We see that 611 of the variability in repair time has been explained by the repair person that performed the service an acceptable but not good fit
36 a The Minitab output is shown below
The regression equation is
Time 186 + 0291 Months + 110 Type 0609 Person
Predictor Coef SE Coef T P
Constant 18602 07286 255 0043
Months 029144 008360 349 0013
Type 11024 03033 363 0011
Person 06091 03879 157 0167
S 04174 RSq 900 RSq(adj) 850
Analysis of Variance
Source DF SS MS F P
Regression 3 94305 31435 1804 0002
Residual Error 6 10455 01743
Total 9 104760
b Since the pvalue corresponding to F 1804 is 002 < a 05 the overall model is statistically significant
c The pvalue corresponding to t 157 is 167 > a 05 thus the addition of Person is not statistically significant Person is highly correlated with Months (the sample correlation coefficient is 691) thus once the effect of Months has been accounted for Person will not add much to the model
37 a Let Position 0 if a guard
Position 1 if an offensive tackle
b The Minitab output is shown below
The regression equation is
Rating 112 + 0732 Position + 00222 Weight 228 Speed
Predictor Coef SE Coef T P
Constant 11223 4523 248 0022
Position 07324 02893 253 0019
Weight 002219 001039 214 0045
Speed 22775 09290 245 0023
S 06936 RSq 475 RSq(adj) 401
Analysis of Variance
Source DF SS MS F P
Regression 3 91562 30521 635 0003
Residual Error 21 101014 04810
Total 24 192576
c Since the pvalue corresponding to F 635 is 003 < a 05 there is a significant relationship between rating and the independent variables
d The value of RSq (adj) is 401 the estimated regression equation did not provide a very good fit
e Since the pvalue for Position is t 253 < a 05 position is a significant factor in the player’s rating
f
38 a The Minitab output is shown below
The regression equation is
Risk 918 + 108 Age + 0252 Pressure + 874 Smoker
Predictor Coef SE Coef T P
Constant 9176 1522 603 0000
Age 10767 01660 649 0000
Pressure 025181 004523 557 0000
Smoker 8740 3001 291 0010
S 5757 RSq 873 RSq(adj) 850
Analysis of Variance
Source DF SS MS F P
Regression 3 36607 12202 3682 0000
Residual Error 16 5302 331
Total 19 41909
b Since the pvalue corresponding to t 291 is 010 < a 05 smoking is a significant factor
c Using Minitab the point estimate is 3427 the 95 prediction interval is 2135 to 4718 Thus the probability of a stroke (2135 to 4718 at the 95 confidence level) appears to be quite high The physician would probably recommend that Art quit smoking and begin some type of treatment designed to reduce his blood pressure
39 a The Minitab output is shown below
The regression equation is
Y 020 + 260 X
Predictor Coef SE Coef T P
Constant 0200 2132 009 0931
X 26000 06429 404 0027
S 2033 RSq 845 RSq(adj) 793
Analysis of Variance
Source DF SS MS F P
Regression 1 67600 67600 1635 0027
Residual Error 3 12400 4133
Total 4 80000
b Using Minitab we obtained the following values
xi
yi
Standardized Residual
1
3
28
16
2
7
54
94
3
5
80
165
4
11
106
24
5
14
132
62
The point (35) does not appear to follow the trend of remaining data however the value of the standardized residual for this point 165 is not large enough for us to conclude that (3 5) is an outlier
c Using Minitab we obtained the following values
xi
yi
Studentized
Deleted Residual
1
3
13
2
7
91
3
5
442
4
11
19
5
14
54
t025 4303 (n p 2 5 1 2 2 degrees of freedom)
Since the studentized deleted residual for (3 5) is 442 < 4303 we conclude that the 3rd observation is an outlier
40 a The Minitab output is shown below
The regression equation is
Y 533 + 311 X
Predicator
Coef
Stdev
tratio
p
Constant
53280
5786
921
0003
X
31100
02016
1543
0001
s 2851 Rsq 988 Rsq (adj) 983
Analysis of Variance
SOURCE
DF
SS
MS
F
p
Regression
1
19344
19344
23803
0001
Error
3
244
81
Total
4
15988
b Using the Minitab we obtained the following values
xi
yi
Studentized
Deleted Residual
22
12
194
24
21
12
26
31
179
28
35
40
40
70
190
t025 4303 (n p 2 5 1 2 2 degrees of freedom)
Since none of the studentized deleted residuals are less than 4303 or greater than 4303 none of the observations can be classified as an outlier
c Using Minitab we obtained the following values
xi
yi
hi
22
12
38
24
21
28
26
31
22
28
35
20
40
70
92
The critical value is
Since none of the values exceed 12 we conclude that there are no influential observations in the data
d Using Minitab we obtained the following values
xi
yi
Di
22
12
60
24
21
00
26
31
26
28
35
03
40
70
1109
Since D5 1109 > 1 (rule of thumb critical value) we conclude that the fifth observation is influential
41 a The Minitab output appears in the solution to part (b) of Exercise 5 the estimated regression equation is
Revenue 832 + 229 TVAdv + 130 NewsAdv
b Using Minitab we obtained the following values
Standardized Residual
9663
162
9041
108
9434
122
9221
37
9439
110
9424
40
9442
112
9335
108
With the relatively few observations it is difficult to determine if any of the assumptions regarding the error term have been violated For instance an argument could be made that there does not appear to be any pattern in the plot alternatively an argument could be made that there is a curvilinear pattern in the plot
c The values of the standardized residuals are greater than 2 and less than +2 thus using test there are no outliers As a further check for outliers we used Minitab to compute the following studentized deleted residuals
Observation
Studentized Deleted Residual
1
211
2
110
3
131
4
33
5
113
6
36
7
116
8
110
t025 2776 (n p 2 8 2 2 4 degrees of freedom)
Since none of the studentized deleted residuals is less tan 2776 or greater than 2776 we conclude that there are no outliers in the data
d Using Minitab we obtained the following values
Observation
hi
Di
1
63
152
2
65
70
3
30
22
4
23
01
5
26
14
6
14
01
7
66
81
8
13
06
The critical average value is
Since none of the values exceed 1125 we conclude that there are no influential observations
However using Cook’s distance measure we see that D1 > 1 (rule of thumb critical value) thus we conclude the first observation is influential Final Conclusion observations 1 is an influential observation
42 a The Minitab output is shown below
The regression equation is
Speed 713 + 0107 Price + 00845 Horsepwr
Predictor Coef SE Coef T P
Constant 71328 2248 3173 0000
Price 010719 003918 274 0017
Horsepwr 0084496 0009306 908 0000
S 2485 RSq 919 RSq(adj) 907
Analysis of Variance
Source DF SS MS F P
Regression 2 91566 45783 7412 0000
Residual Error 13 8030 618
Total 15 99595
Source DF Seq SS
Price 1 40639
Horsepwr 1 50927
Unusual Observations
Obs Price Speed Fit SE Fit Residual St Resid
2 938 108000 105882 2007 2118 145 X
X denotes an observation whose X value gives it large influence
b The standardized residual plot is shown below There appears to be a very unusual trend in the standardized residuals
xx x
12+
x
SRES1 x
x
x
00+ x x
x
x x x
12+ x
x
x
+++++FITS1
900 960 1020 1080 1140
c The Minitab output shown in part (a) did not identify any observations with a large standardized residual thus there does not appear to be any outliers in the data
d The Minitab output shown in part (a) identifies observation 2 as an influential observation
43 a The Minitab output is shown below
The regression equation is
College 266 + 00970 SatScore
Predictor Coef SE Coef T P
Constant 2661 3722 072 0485
SatScore 009703 003734 260 0019
S 1283 RSq 297 RSq(adj) 253
Analysis of Variance
Source DF SS MS F P
Regression 1 11108 11108 675 0019
Residual Error 16 26323 1645
Total 17 37431
Unusual Observations
Obs SatScore College Fit SE Fit Residual St Resid
3 716 4000 4286 1079 286 041 X
X denotes an observation whose X value gives it large influence
b The Minitab output shown in part a identifies observation 3 as an influential observation
c The Minitab output appears in the solution to Exercise 9 the estimates regression equation is College 267 143 Size + 00757 SATScore
d The following Minitab output was also provided as part of the regression output for part c
Unusual Observations
Obs Size College Fit StdevFit Residual StResid
3 300 400 3804 1097 196 034 X
X denotes an obs whose X value gives it large influence
Observation 3 is still identified as an influential observation
44 a The expected increase in final college grade point average corresponding to a one point increase in high school grade point average is 0235 when SAT mathematics score does not change Similarly the expected increase in final college grade point average corresponding to a one point increase in the SAT mathematics score is 00486 when the high school grade point average does not change
b 141 + 0235(84) + 00486(540) 319
45 a Job satisfaction can be expected to decrease by 869 units with a one unit increase in length of service if the wage rate does not change A dollar increase in the wage rate is associated with a 135 point increase in the job satisfaction score when the length of service does not change
b 144 869(4) + 135(65) 6739
46 a The computer output with the missing values filled in is as follows
The regression equation is
Y 8103 + 7602 X1 + 3111 X2
Predicator
Coef
Stdev
tratio
Constant
8103
2667
304
X1
7602
2105
361
X2
3111
0613
508
s 335 Rsq 923 Rsq (adj) 910
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
1612
806
7182
Error
12
13467
112225
Total
14
174667
b t025 2179 (12 DF)
for b1 361 > 2179 reject H0 b1 0
for b2 508 > 2179 reject H0 b2 0
c See computer output
d
47 a The regression equation is
Y 141 + 0235 X1 + 00486 X2
Predictor
Coef
Stdev
tratio
Constant
14053
04848
290
X1
0023467
0008666
271
X2
00486
0001077
451
s 01298 Rsq 937 Rsq (adj) 919
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
176209
881
5244
Error
7
1179
0168
Total
9
188000
b F05 474 (2 DF numerator 7 DF denominator)
F 5244 > F05 significant relationship
c
good fit
d t025 2365 (7 DF)
for B1 t 271 > 2365 reject H0 B1 0
for B2 t 451 > 2365 reject H0 B2 0
48 a The regression equation is
Y 144 869 X1 + 1352 X2
Predictor
Coef
Stdev
tratio
Constant
14448
8191
176
X1
869
1555
559
X2
13517
2085
648
s 3773 Rsq 901 Rsq (adj) 861
Analysis of Variance
SOURCE
DF
SS
MS
F
Regression
2
64883
324415
2279
Error
5
7117
14234
Total
7
72000
b F05 579 (5 DF)
F 2279 > F05 significant relationship
c
good fit
d t025 2571 (5 DF)
for b1 t 559 < 2571 reject H0 b1 0
for b2 t 648 > 2571 reject H0 b2 0
49 a The Minitab output is shown below
The regression equation is
Price 128 + 226 BookVal
Predictor Coef SE Coef T P
Constant 12793 6624 193 0064
BookVal 22649 06631 342 0002
S 1950 RSq 294 RSq(adj) 269
Analysis of Variance
Source DF SS MS F P
Regression 1 44339 44339 1167 0002
Error 28 106423 3801
Total 29 150761
b The value of Rsq is 294 the estimated regression equation does not provide a good fit
c The Minitab output is shown below
The regression equation is
Price 588 + 254 BookVal + 0484 ReturnEq
Predictor Coef SE Coef T P
Constant 5877 5545 106 0299
BookVal 25356 05331 476 0000
ReturnEq 04841 01174 412 0000
S 1555 RSq 567 RSq(adj) 535
Analysis of Variance
Source DF SS MS F P
Regression 2 85442 42721 1766 0000
Error 27 65319 2419
Total 29 150761
Since the pvalue corresponding to the F test is 0000 the relationship is significant
50 a The Minitab output is shown below
The regression equation is
Speed 976 + 00693 Price 000082 Weight + 00590
Horsepwr 248 Zero60
Predictor Coef SE Coef T P
Constant 9757 1179 827 0000
Price 006928 003805 182 0096
Weight 0000816 0002593 031 0759
Horsepwr 005901 001543 382 0003
Zero60 24836 09601 259 0025
S 2127 RSq 950 RSq(adj) 932
Analysis of Variance
Source DF SS MS F P
Regression 4 94618 23655 5228 0000
Residual Error 11 4977 452
Total 15 99595
b Since the pvalue corresponding to the F test is 0000 the relationship is significant
c Since the pvalues corresponding to the t test for both Horsepwr (pvalue 003) and Zero60 (pvalue 025) are less than 05 both of these independent variables are significant
d The Minitab output is shown below
The regression equation is
Speed 103 + 00558 Horsepwr 319 Zero60
Predictor Coef SE Coef T P
Constant 103103 9448 1091 0000
Horsepwr 005582 001452 384 0002
Zero60 31876 09658 330 0006
S 2301 RSq 931 RSq(adj) 920
Analysis of Variance
Source DF SS MS F P
Regression 2 92712 46356 8754 0000
Residual Error 13 6884 530
Total 15 99595
Source DF Seq SS
Horsepwr 1 86943
Zero60 1 5768
Unusual Observations
Obs Horsepwr Speed Fit SE Fit Residual St Resid
2 290 108000 103352 1015 4648 225R
12 155 84600 82747 1773 1853 126 X
R denotes an observation with a large standardized residual
X denotes an observation whose X value gives it large influence
e The standardized residual plot is shown below
SRES x
15+
x x
2 x x
00+ x x 2
x
xx
15+
x x
++++++FIT
840 900 960 1020 1080 1140
There is an unusual trend in the plot and one observation appears to be an outlier
f The Minitab output indicates that observation 2 is an outlier
g The Minitab output indicates that observation 12 is an influential observation
51 a The Minitab output is shown below
640+
x
Exposure
480+
x
x
320+
x
160+ x 3 x
x
++++++TimesAir
15 30 45 60 75 90
b The Minitab output is shown below
The regression equation is
Exposure 532 + 674 TimesAir
Predictor Coef SE Coef T P
Constant 5324 1653 322 0012
TimesAir 67427 04472 1508 0000
S 3170 RSq 966 RSq(adj) 962
Analysis of Variance
Source DF SS MS F P
Regression 1 228520 228520 22736 0000
Error 8 8041 1005
Total 9 236561
Since the pvalue is 0000 the relationship is significant
c The Minitab output is shown below
The regression equation is
Exposure 731 + 504 TimesAir + 101 BigAds
Predictor Coef SE Coef T P
Constant 73063 7507 973 0000
TimesAir 50368 03268 1541 0000
BigAds 10111 1599 632 0000
S 1308 RSq 995 RSq(adj) 993
Analysis of Variance
Source DF SS MS F P
Regression 2 235363 117682 68784 0000
Error 7 1198 171
Total 9 236561
d The pvalue corresponding to the t test for BigAds is 0000 thus the dummy variable is significant
e The dummy variable enables us to fit two different lines to the data this approach is referred to as piecewise linear approximation
52 a The Minitab output is shown below
Resale 388 +0000766 Price
Predictor Coef SE Coef T P
Constant 38772 4348 892 0000
Price 00007656 00001900 403 0000
S 5421 RSq 367 RSq(adj) 344
Analysis of Variance
Source DF SS MS F P
Regression 1 47725 47725 1624 0000
Residual Error 28 82292 2939
Total 29 130017
Since the pvalue corresponding to F 1624 is 000 < a 05 there is a significant relationship between Resale and Price
b RSq 367 not a very good fit
c Let Type1 0 and Type2 0 if a small pickup Type1 1 and Type2 0 if a fullsize pickup and Type1 0 and Type2 1 if a sport utility
The Minitab output using Type1 Type2 and Price is shown below
The regression equation is
Resale 426 + 909 Type1 + 792 Type2 +0000341 Price
Predictor Coef SE Coef T P
Constant 42554 3562 1195 0000
Type1 9090 2248 404 0000
Type2 7917 2163 366 0001
Price 00003415 00001800 190 0069
S 4298 RSq 631 RSq(adj) 588
Analysis of Variance
Source DF SS MS F P
Regression 3 81977 27326 1479 0000
Residual Error 26 48040 1848
Total 29 130017
d Since the pvalue corresponding to F 1479 is 000 < a 05 there is a significant relationship between Resale and the independent variables Note that individually Price is not significant at the 05 level of significance If we rerun the regression using just Type1 and Type2 the value of RSq (adj) decreases to 544 a drop of only 4 Thus it appears that for these data the type of vehicle is the strongest predictor of the resale value
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