Source
Data from
McClave and Benson, Statistics for Business and Economics 8th ed. Exercise 10.79, p. 529
Earlier ed., Exercise 11.77
15 shipments of flour bags with a nominal weight of 50 lbs.
Worksheets:
Codebook
Data
with intercept
p=0
p=1
p=2
Descriptive Stats
Exercises
Shipment Weight No_Bags
1 5050 100
2 10249 205
3 20000 450
4 7420 150
5 24685 500
6 10206 200
7 7325 150
8 4958 100
9 7162 150
10 24000 500
11 4900 100
12 14501 300
13 28000 600
14 17002 400
15 16100 400
MSB8/e, Exercise 10.79, p. 529: Test whether an intercept is needed in the model.
Shipment Weight No_Bags SUMMARY OUTPUT
1 5050 100
2 10249 205 Regression Statistics
3 20000 450 Multiple R 0.99
4 7420 150 R Square 0.98
5 24685 500 Adjusted R Square 0.98
6 10206 200 Standard Error 1027.29
7 7325 150 Observations 15
8 4958 100
9 7162 150 ANOVA Significance
10 24000 500 df SS MS F F
11 4900 100 Regression 1 849116841.2 849116841.2 804.6 4.45E-13
12 14501 300 Residual 13 13719201.2 1055323.2
13 28000 600 Total 14 862836042.4
14 17002 400
15 16100 400 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 90.0% Upper 90.0%
Intercept 478.4 528.3 0.9 0.3816 -662.8 1619.7 -457.1 1414.0
No_Bags 45.2 1.6 28.4 0.0000 41.7 48.6 42.3 48.0
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.99
R Square 0.98
Adjusted R Square 0.91
Standard Error 1020.67
Observations 15
ANOVA
df SS MS F Significance F
Regression 1 848251195.37 848251195.37 814.24 4.1271E-13
Residual 14 14584847.03 1041774.79
Total 15 862836042.40
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 90.0% Upper 90.0%
Intercept 0.00 #N/A #N/A #N/A #N/A #N/A #N/A #N/A
No_Bags 46.40 0.79 58.43 0.00 44.70 48.10 45.00 47.80
Criterion is (n/2)lnRSS(p) + pSUM(ln xi)
RSS= 14584847.03
lnRSS: 16.50
SUM(ln xi): 82.09084879
Criterion: 123.72 <162<201
MLE of p is p = 0 (OLS).
Shipment Weight No_Bags LN xi y(1) x(1)
1 5050 100 4.6051702 505.00 10.00
2 10249 205 5.32301 715.82 14.32
3 20000 450 6.1092476 942.81 21.21
4 7420 150 5.0106353 605.84 12.25
5 24685 500 6.2146081 1103.95 22.36
6 10206 200 5.2983174 721.67 14.14
7 7325 150 5.0106353 598.08 12.25
8 4958 100 4.6051702 495.80 10.00
9 7162 150 5.0106353 584.77 12.25
10 24000 500 6.2146081 1073.31 22.36
11 4900 100 4.6051702 490.00 10.00
12 14501 300 5.7037825 837.22 17.32
13 28000 600 6.3969297 1143.10 24.49
14 17002 400 5.9914645 850.10 20.00
15 16100 400 5.9914645 805.00 20.00
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.969799
R Square 0.94
Adjusted R Square 0.87
Standard Error 54.70
Observations 15
ANOVA
df SS MS F Significance F
Regression 1 662338.55 662338.55 221.33 1.53E-09
Residual 14 41895.06 2992.50
Total 15 704233.62
CoefficientsStandard Error t Stat P-value Lower 95% Upper 90.0%
Upper 95% Lower 90.0%
Intercept 0 #N/A #N/A #N/A #N/A #N/A #N/A #N/A
x(1) 46.82 0.83 56.16 0.00 45.03 48.61 45.35 48.29
RSS= 41895.06
Criterion is (n/2)lnRSS(p) + pSUM(ln xi)
lnRSS: 10.64292331
SUM(ln xi): 82.09084879
Criterion: 161.9127736
200.7609165 for p of 2
Shipment Weight No_Bags Ratio LN xi Column1
1 5050 100 50.50 4.60517
2 10249 205 50.00 5.32301 Mean 47.71497109
3 20000 450 44.44 6.10925 Standard Error 0.790618942
4 7420 150 49.47 5.01064 Median 48.83333333
5 24685 500 49.37 6.21461 Mode #N/A
6 10206 200 51.03 5.29832 Standard Deviation 3.062053994
7 7325 150 48.83 5.01064 Sample Variance 9.376174664
8 4958 100 49.58 4.60517 Kurtosis 1.479726398
9 7162 150 47.75 5.01064 Skewness -1.44531319
10 24000 500 48.00 6.21461 Range 10.78
11 4900 100 49.00 4.60517 Minimum 40.25
12 14501 300 48.34 5.70378 Maximum 51.03
13 28000 600 46.67 6.39693 Sum 715.7245664
14 17002 400 42.51 5.99146 Count 15
15 16100 400 40.25 5.99146
RSS= 131.2664453
Criterion is (n/2)lnRSS(p) + pSUM(ln xi)
lnRSS: 4.877229191
SUM(ln xi): 82.09084879
Criterion: 200.7609165
Shipment Weight No_Bags Column1
1 5050 100 50.50
2 10249 205 50.00 Mean 47.71
3 20000 450 44.44 Standard Error 0.79 -2.89
4 7420 150 49.47 Median 48.83
5 24685 500 49.37 Mode #N/A
6 10206 200 51.03 Standard Deviation 3.06
7 7325 150 48.83 Sample Variance 9.38
8 4958 100 49.58 Kurtosis 1.48
9 7162 150 47.75 Skewness -1.45
10 24000 500 48.00 Range 10.78
11 4900 100 49.00 Minimum 40.25
12 14501 300 48.34 Maximum 51.03
13 28000 600 46.67 Sum 715.725
14 17002 400 42.51 Count 15
15 16100 400 40.25
sum: 201558 4305 46.82
1. Compute the 15 ratios, weight/No_Bags
2. Compute the mean of the ratios.
3. Compute the ratio of the means.
4. Compute the least squares estimate of the average bag-weight.
5. Find the formula for the weighted least-squares estimate if Var(yi) = s2 xip.