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Unformatted text preview: SOLUTION MA206 Suggested Problems
Lessona Minimizing Sum of Squares of a Line 9 1. Given the following sample ot‘pushups and run times from the APFT, develop a linear
regression model that will predict a soldier’s run time (in seconds) from the number ot‘pushups
that he did. Pushups 40 40 53 55 50 66 79 78 70 86 90 _ 66 60 88 RunTime(see) 1056"“103?“ 887 "7239— 954 782 840 778 855 881 865 l()l3_j§_5 896 " a. Create a scatter plot of the data. Does it seem sensible to model this data with a linear
equation? 1100
1050 Run Time (sec)
;
CD (0 (O O
0'! O 01 O
O O O 0
O6
0
9
¢
G
0
¢ Pushups It appears that there may be a useful linear relationship between Pushups and Run Time. h. Develop a linear model by ﬁnding values for m and b that minimize the SSE. Add the
linear model to your graph. Comment on the adequacy of your linear model. 30 40 5O 60 7O 80 90 100 Pushups y = ~3.089x + 1099.27
SSE = 72,036 SOLUTION The model seems to capture the overall trend of the data set reasonably well, but the very
large SSE value indicates that a linear model is not the best choice for this data set. c. Using your model, predict the run times for soldiers who completed 62 and 99 pushups.
Are these predictions valid? Why or why not? When x = 62, )3 : —3.089(62) + 1099.27 = 907.76 see. This prediction is mathematically correct according to our model, but cannot be completely relied upon because a linear model is not a
great choice for this data set. V varlinfi n11 n rr ‘Iqjili 1"!) 1
vcu u UM on v
VCLUOU A C) :00 0:_’2 S20 0(\4‘1
// .1. //II 7 y \lU/\/ CD Z 06C. plvxtiuuui 5.... r
99 is outside the range of the x—va ues in our original data set. The danger of extrapolation is
that even if we had a very strong linear relationship here (which we don’t), the linear relationship
may not be valid for such xavalues. 2. A warehouse manager is interested in possible improvements to labor efﬁciency if air
conditioning is installed in his warehouse. He has provided you with the following sample data: “Temp (F) 52 68 64 88 80 75 59 63 85 74 71 l 66
Unloading Time (min) “64 53 58 59 49 54 38 48 68 63 58 L47 a. Create a scatter plot of the data. Does it seem sensible to model this data with a linear equation?
80
E 70 .
i=1 ‘ s
g 60 . . o
i: ‘ *
g, 50 , . c
.5
:3 4o .
5
30
2O ‘ ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ., ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,1 ,,,,,,,,,,,,,,,,,,,,,,, a
40 50 so 70 so 90 100
Temp (F) It appears that there might be a useful linear relationship between Temp and Unloading
Time. h. Develop a linear model by ﬁnding values for m and b that minimize the SSE. Add the
linear model to your graph. Comment on the adequacy of your linear model. SOLUTION 80
E 70 9
E ° ~
«E: 60 WWWW A WWW e A 9 °
5: . “W “WW M w» w» WWW W 50 ' ‘ °
1:
a 40 .
'E
D 30
40 50 60 7O 80 90 1 00
Tnmn IF\
...,, ‘ ,
)9 = 0.265936 + 36.19 SSE = 703.33, which indicates that a linear model is a poor ﬁt for this data set.
c. Provide your recommendation to the warehouse manager. Since there does not seem to be a strong linear relationship between Temp and Unloading
Time, the manger should not install air conditioning in the warehouse. ...
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 Spring '10
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