Question

SUMMARY OUTPUT | ||||||||

Regression Statistics | ||||||||

Multiple R | 0.884651238 | |||||||

R Square | 0.782607814 | |||||||

Adjusted R Square | 0.601447658 | |||||||

Standard Error | 25.32612538 | |||||||

Observations | 12 | |||||||

ANOVA | ||||||||

df | SS | MS | F | Significance F | ||||

Regression | 5 | 13854.44091 | 2770.888181 | 4.319977601 | 0.051673038 | |||

Residual | 6 | 3848.475761 | 641.4126268 | |||||

Total | 11 | 17702.91667 | ||||||

Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |

Intercept | -53.17436031 | 42.95203957 | -1.237993838 | 0.261960445 | -158.274215 | 51.92549434 | -158.274215 | 51.92549434 |

Advertising ($1000s) | 2.050813091 | 0.763960482 | 2.684449181 | 0.036320193 | 0.181469133 | 3.92015705 | 0.181469133 | 3.92015705 |

t (quarters) | -4.047065728 | 2.779316427 | -1.456137088 | 0.19560701 | -10.84780803 | 2.753676575 | -10.84780803 | 2.753676575 |

Q1 | 19.42140471 | 21.88478307 | 0.887438758 | 0.409003775 | -34.12873036 | 72.97153977 | -34.12873036 | 72.97153977 |

Q2 | 23.03418679 | 27.39517297 | 0.840811876 | 0.432677603 | -43.99938661 | 90.06776019 | -43.99938661 | 90.06776019 |

Q3 | 20.943922 | 25.53508827 | 0.820201668 | 0.443457241 | -41.5381881 | 83.4260321 | -41.5381881 | 83.4260321 |

1) Is there constant seasonality; increasing/decreasing seasonality; a trend; or no time effect at all from the linear regression analysis shown above? And why?

2) Are models with seasonality (Q1, Q2, Q3 included) better than models without? Why?

Answer #1

we can form the regression equation based on the coefficients as

Y = -53.17 +2.05*advertsing -4.04*t + 19.42*q1 +23.03*q2 +20.94*q3

we see that the coeffecient of q2 is higher than that of q1 and q3
, which means in q2 the sale is effected by 23.03 units , hence
apparently there appears to be a seasonality , with high sales in
q2

for the second part , we simply compare the r2 value of the seasonal model with the r2 value of the non seasonal model, if this value is less than 0.7826 (r2 of seasonal model) then seasonal model is better else non seasonal model would be better. However , without the data we cant run the regression

SUMMARY OUTPUT Regression Statistics Multiple R 0.84508179 R
Square 0.714163232 Adjusted R Square 0.704942691 Standard Error
9.187149383 Observations 33 ANOVA df SS MS F Significance F
Regression 1 6537.363661 6537.363661 77.4535073 6.17395E-10
Residual 31 2616.515127 84.40371378 Total 32 9153.878788
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Lower 95.0% Upper 95.0% Intercept 61.07492285 3.406335763
17.92980114 6.41286E-18 54.12765526 68.02219044 54.12765526
68.02219044 Time (Y) -0.038369095 0.004359744 -8.800767426
6.17395E-10 -0.047260852 -0.029477338 -0.047260852 -0.029477338
Using your highlighted cells, what is the equation...

SUMMARY OUTPUT
Regression Statistics
Multiple R
0.993709623
R Square
0.987458816
Adjusted R Square
0.987378251
Standard Error
514.2440271
Observations
471
ANOVA
df
SS
MS
F
Significance F
Regression
3
9723795745
3241265248
12256.7707
0
Residual
467
123496711.4
264446.9194
Total
470
9847292456
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
-267.1127974
42.01832073
-6.357055513
4.8988E-10
-349.68118
-184.54441
-349.68118
-184.54441
Fuel cost (000,000)
0.449917223
0.098292092
4.577349137
6.0451E-06
0.25676768
0.64306676
0.25676768
0.64306676
Salary (000,000)
-0.327915884
0.188252958
-1.741889678
0.08218614
-0.6978436...

SUMMARY OUTPUT
Regression Statistics
Multiple R
0.909785963
R
Square
0.827710499
Adjusted R Square
0.826591736
Standard Error
7.177298036
Observations
156
ANOVA
df
SS
MS
F
Significance F
Regression
1
38112.05194
38112.05194
739.8443652
1.09619E-60
Residual
154
7933.095493
51.5136071
Total
155
46045.14744
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
8.67422449
2.447697434
3.543830365
0.000522385
3.838827439
13.50962154
3.838827439
13.50962154
X
Variable 1
0.801382837
0.029462517
27.20008024
1.09619E-60
0.743179986
0.859585688
0.743179986
0.859585688
(d)
How much of the variation in...

Regression Statistics
Multiple
R
0.710723
R
Square
0.505127
Adjusted
R Square
0.450141
Standard
Error
1.216847
Observations
21
ANOVA
df
SS
MS
F
Significance F
Regression
2
27.20518
13.60259
9.186487
0.00178
Residual
18
26.65291
1.480717
Total
20
53.8581
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
58.74307
12.66908
4.636728
0.000205
32.12632
85.35982
32.12632
85.35982
High
School Grad
-0.00133
0.000311
-4.28236
0.000448
-0.00198
-0.00068
-0.00198
-0.00068
Bachelor's
-0.00016
5.46E-05
-3.00144
0.007661
-0.00028
-4.9E-05
-0.00028
-4.9E-05...

Regression Statistics
Multiple
R
0.3641
R
Square
0.1325
Adjusted
R Square
0.1176
Standard
Error
0.0834
Observations
60
ANOVA
df
SS
MS
F
Significance F
Regression
1
0.0617
0.0617
8.8622
0.0042
Residual
58
0.4038
0.0070
Total
59
0.4655
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
-0.0144
0.0110
-1.3062
0.1966
-0.0364
0.0077
X
Variable 1
0.8554
0.2874
2.9769
0.0042
0.2802
1.4307
How do you interpret the above table?

SUMMARY OUTPUT Regression Statistics Multiple R 0.440902923 R
Square 0.194395388 Adjusted R Square 0.165100675 Standard Error
0.428710255 Observations 115 ANOVA df SS MS F Significance F
Regression 4 4.878479035 1.219619759 6.635852231 8.02761E-05
Residual 110 20.21717314 0.183792483 Total 114 25.09565217
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Lower 95.0% Upper 95.0% Intercept 0.321875686 0.323939655
0.99362854 0.322584465 -0.320096675 0.963848047 -0.320096675
0.963848047 Gender -0.307211858 0.082630734 -3.717888514
0.000317832 -0.470966578 -0.143457137 -0.470966578 -0.143457137 Age
0.000724105 0.091134233 0.007945479 0.993674883 -0.179882553
0.181330763 -0.179882553 0.181330763...

SUMMARY OUTPUT
Regression Statistics
Multiple R
0.870402
R
Square
0.7576
Adjusted R Square
0.68488
Standard Error
1816.52
Observations
27
ANOVA
df
SS
MS
F
Significance F
Regression
6
2.06E+08
34376848
10.41804
2.81E-05
Residual
20
65994862
3299743
Total
26
2.72E+08
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
-4695.4
12622.97
-0.37197
0.713825
-31026.5
21635.66
-31026.5
21635.66
AGE
161.7028
126.5655
1.277621
0.216015
-102.308
425.7137
-102.308
425.7137
MILAGE
-0.03441
0.023186
-1.4842
0.153346
-0.08278
0.013953
-0.08278
0.013953...

Dep.=
Mileage
Indep.=
Cylinders
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
7.0000
ANOVA
Significance
df
SS
MS
F
F
Regression
12.4926
Residual
Total
169.4286
Standard
Coefficients
Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
38.7857
Cylinders
-2.7500
SE
CI
CI
PI
PI
Predicted
Predicted
Lower
Upper
Lower
Upper
x0
Value
Value
95%
95%
95%
95%
4.0000
1.9507
6.0000
1.1763
Is there a relationship between a car's gas
MILEAGE (in miles/gallon) and its...

Dep.=
Mileage
Indep.=
Length
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
7.0000
ANOVA
Significance
df
SS
MS
F
F
Regression
6.1135
Residual
Total
169.4286
Standard
Coefficients
Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
80.0094
Length
-0.3047
SE
CI
CI
PI
PI
Predicted
Predicted
Lower
Upper
Lower
Upper
x0
Value
Value
95%
95%
95%
95%
175.0000
2.3108
210.0000
2.9335
Is there a relationship between a car's gas
MILEAGE (in miles/gallon) and its...

SUMMARY OUTPUT
Regression Statistics
Multiple
R
0.231960777
R
Square
0.053805802
Adjusted
R Square
0.034093423
Standard
Error
5272.980333
Observations
50
ANOVA
df
SS
MS
F
Significance F
Regression
1
75893113.09
75893113.09
2.729543781
0.105035125
Residual
48
1334607437
27804321.59
Total
49
1410500550
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 99.0%
Upper 99.0%
Intercept
6396.894057
3281.342486
1.949474669
0.057094351
-200.6871963
12994.47531
-2404.335972
15198.12409
HSRANK
64.68225855
39.15075519
1.6521331
0.105035125
-14.03561063
143.4001277
-40.32805468
169.6925718
a. According to your estimate, what is the predicted...

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