Chapter 18
18.89
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.4394
R Square
0.1931
Adjusted R Square
0.1258
Standard Error
0.0567
Observations
14
ANOVA
df
Regression
Residual
Total
Intercept
Team BA
1
12
13
SS
0.0092
0.0386
0.0478
Coefficients Standard Error
-0.227
0.43
2.794
1.65
MS
0.0092
0.0032
F
2.87
Significance F
0.1160
t Stat
P-value
-0.53
0.6069
1.69
0.1160
a ŷ = -.227 + 2.794x. The slope is 2.794; for each additional point increase in batting average, the
team’s winning percentage increases on average by 2.794 points.
b s = .0567. This statistic is large relative to the average winning percentage, .500. The model is
poor.
c t = 1.69, p-value = .1160/2 = .0580; there is not enough evidence to infer a positive linear
relationship between team batting average and winning percentage.
d R 2 = .1931; 19.31% of the variation in winning percentage is explained by the variation in team
batting average.
e
Prediction Interval
Winning%
Predicted value
0.542
Prediction Interval
Lower limit
Upper limit
0.428
0.655
Interval Estimate of Expected Value
Lower limit
0.490
Upper limit
0.593
Lower prediction limit = .428, Upper prediction limit = .655.
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18.94
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.2248
R Square
0.0505
Adjusted R Square
0.0467
Standard Error
8.28
Observations
250
ANOVA
df
Regression
Residual
Total
Intercept
Height
1
248
249
SS
905.60
17010.97
17916.56
Coefficients Standard Error
17.93
11.48
0.604
0.166
MS
905.60
68.59
F
Significance F
13.20
0.0003
t Stat
P-value
1.56
0.1194
3.63
0.0003
ŷ = 17.93 + .60x. The slope is .60; for each additional inch of height, annual income increases on
average by .60 thousand dollars ($600).
b t = 3.63, p-value = .0003/2 = .0002. There is enough evidence to infer a positive linear relationship
between height and income.
c R 2 = .0505; 5.05% of the variation in incomes is explained by the variation in heights.
d The model is too poor to be used to predict or estimate.
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Chapter 19
19.2
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.8734
R Square
0.7629
Adjusted R Square
0.7453
Standard Error
3.75
Observations
30
ANOVA
df
Regression
Residual
Total
Intercept
Assignment
Midterm
2
27
29
SS
1223.18
380.18
1603.37
Coefficients Standard Error
13.01
3.528
0.194
0.200
1.112
0.122
MS
611.59
14.08
F
Significance F
43.43
0.0000
t Stat
P-value
3.69
0.0010
0.97
0.3417
9.12
0.0000
a yˆ  13.01  .194 x1  1.112 x2
b The standard error of estimate is s = 3.75. It is an estimate of the standard
deviation of the error variable.
c The coefficient of determination is R 2 = .7629; 76.29% of the variation in final
exam marks is explained by the model.
d The coefficient of determination adjusted for degrees of freedom is .7453. It differs
from R 2 because it includes an adjustment for the number of independent variables.
e
H 0 : 1   2  0
H 1 : At least one  i is not equal to zero
F = 43.43, p-value = 0. There is enough evidence to conclude that the model is valid.
f b1 = .194; for each addition mark on assignments the final exam mark on average
increases by .194 provided that the other variable remains constant.
b2 = 1.112; for each addition midterm mark the final exam mark on average increases
by 1.112 provided that the other variable remains constant.
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g
H 0 : 1  0
H 1 : 1  0
t = .97, p-value = .3417. There is not enough evidence to infer that assignment marks
and final exam marks are linearly related.
h
H 0 : 2  0
H1 : 2  0
t = 9.12, p-value = 0. There is sufficient evidence to infer that midterm marks and
final exam marks are linearly related.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.4419
R Square
0.1953
Adjusted R Square
0.0803
Standard Error
2.59
Observations
25
ANOVA
df
Regression
Residual
Total
Intercept
Direct
Newspaper
Television
SS
3
21
24
34.10
140.56
174.66
Coefficients Standard Error
12.31
4.70
0.57
1.72
3.32
1.54
0.73
1.96
MS
11.37
6.69
F
1.70
Significance F
0.1979
t Stat
P-value
2.62
0.0160
0.33
0.7437
2.16
0.0427
0.37
0.7123
19.7
a The regression equation is yˆ  12.31  .57 x1  3.32 x2  .73 x3
b The coefficient of determination is R 2 = .1953; 19.53% of the variation in sales is
explained by the model. The coefficient of determination adjusted for degrees of
freedom is .0803. The model fits poorly.
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c The standard error of estimate is s = 2.59. It is an estimate of the standard
deviation of the error variable.
d
H 0 : 1   2  3  0
H 1 : At least one  i is not equal to zero
F = 1.70, p-value = .1979. There is not enough evidence to conclude that the model
is valid.
e
H 0 : i  0
H 1 : i  0
Direct: t = .33, p-value = .7437
Newspaper: t = 2.16, p-value = .0427
Television: t = .37, p-value = .7123
Only expenditures on newspaper advertising is linearly related to sales.
Prediction Interval
Sales
Predicted value
18.21
Prediction Interval
Lower limit
Upper limit
12.27
24.15
Interval Estimate of Expected Value
Lower limit
15.70
Upper limit
20.73
f&g
f We predict that sales will fall between $12,270 and $24,150.
g We estimate that mean sales will fall between $15,700 and $20,730.
h The interval in part f predicts one week’s gross sales, whereas the interval in part h
estimates the mean weekly gross sales.
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