Linear Regression Worksheet – 2007 NFL Scores/Vegas Forecasts
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Spread (S): Projected point differential: Away Team – Home Team (<0  Home favored)
Over/Under (OU): Projected point total: Away Team + Home Team
Projected Away Score: X1 = (OU+S)/2 Projected Home Score: X1 = (OU-S)/2
Other Possible Predictors:
o Home Field Effect: X2=1 if Home, 0 if Away
o Home Field x Forecast Interaction: X3=X1X2
Response: Y=Actual Score
Regression Model: Yi = 0 + 1Xi1 + 2Xi2 + 3Xi3 + i i=1,…,512 (256 games x 2 teams)
Matrix Form: Y = X + 
Hypothesis that the mean actual score is exactly equal to the Vegas Forecast, with no home team effects:
H0: 
Game
1
2
3
…
254
255
256
1
2
3
…
254
255
256
i
1
2
3
…
254
255
256
257
258
259
…
510
511
512
X0
1
1
1
…
1
1
1
1
1
1
…
1
1
1
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K'  
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sse
mse
49615.22 97.66775
H0: K’-m = 0
Parameter
X1
28
19.25
21.75
…
17.75
26.75
20
21.5
16.75
20.75
…
25.25
21.25
14
X2
1
1
1
…
1
1
1
0
0
0
…
0
0
0
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m
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
s
9.8827
HA: K’-m ≠ 0
Estimate
X3
28
19.25
21.75
…
17.75
26.75
20
0
0
0
…
0
0
0
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Y
41
24
13
…
17
48
13
10
3
27
…
30
19
10
X'X
512
10696
256
5664.5
10696
232904.8
5664.5
129445.4
256
5664.5
256
5664.5
5664.5
129445.4
5664.5
129445.4
X'Y
11104
241207.8
5919
134504.5
(X'X)^-1
0.08846
-0.00430
-0.08846
0.00430
-0.00430
0.00022
0.00430
-0.00022
-0.08846
0.00430
0.21157
-0.00969
0.00430
-0.00022
-0.00969
0.00046
Beta-hat
-0.377
1.050
4.453
-0.189
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^
K ' β m  
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Q(K)
436.033
df(K)
4
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K' (X' X)
K

1

df(E)
508
Test Stat: _________________
Std. Error
1
t-stat
P-value
Rejection Region: __________
95% Confidence Interval
Lower Bound
Upper Bound
INTERCEPT
Vegas Pick (X1)
Home (X2)
Vegas*Home (X3)
Fitted Equation:
Home _________________________________
Away _________________________
Model with only Vegas Pick (X1):
X'X
512
10696
10696
232904.8
X'Y
11104
241207.75
(X'X)^-1
0.04809
-0.00221
-0.00221
0.00011
Beta-hat
1.28
0.98

Compute a 95% Confidence Interval for the mean of all games where the Vegas Pick = 25
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Compute a 95% Prediction Interval for the a single (future) game where the Vegas Pick = 25
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Test whether a linear relation is reasonable fit for this data (F-test for lack of Fit)
# of observations = 512
df
Lack of Fit
Pure Error
Residual
# of distinct levels of X1 = 81
SS
MS
9039
40682
49721
F
(See spreadsheet for data)
F(.05)
P-value