Question

In: Statistics and Probability

Please review the following case information and determine what the population residual plot means. The residual...

Please review the following case information and determine what the population residual plot means. The residual is already calculated but I need to know what role in plays in this case analysis for the regression analysis. Thanks in advance!

In this case, Armand's Pizza Parlors is a chain of Italian-food restaurants located in five-state area. Their most sucessful locations are the ones near college campuses. The null hypothesis is that the managers believe that quarterly sales for these restaurants (y) are related positively to the size of the student population (x). Thus, the restaurants that are near college campuses with a larger student population tend to generate more sales than the restuarants near campuses with a small student population. The alternative hypothesis is that the restaurants near college campuses with a larger student population do not generate more sales than the restaurants near college campuses with a smaller student population.

Restaurant Population Sales
1 15 148
2 22 134
3 2 54
4 13 136
5 13 142
6 8 109
7 10 169
8 19 128
9 3 99
10 24 142
11 25 149
12 10 142
13 15 156
14 22 139
15 8 104
16 17 134
17 15 166
18 23 127
19 14 151
20 3

115

Regression Statistics
Multiple R 0.552066
R Square 0.304777
Adjusted R Square 0.266154
Standard Error 22.57531
Observations 20
ANOVA
df SS MS F Significance F
Regression 1 4021.597 4021.597 7.890983 0.011608
Residual 18 9173.603 509.6446
Total 19 13195.2
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0%
Intercept 103.4275 11.41902 9.057477 4.01E-08 79.43703 127.418 70.55853 136.2965
Population 2.047865 0.729014 2.809089 0.011608 0.516264 3.579466 -0.05056 4.146288

Solutions

Expert Solution

Residual is a more important part of the regression model

using residual plot you can identify the model is the best fit

Because Residual plot must follow the normal distribution

if the residual plot is a straight line then data follows the normality assumption

the residual is the difference between actual and predicted values.

Residual = Actual sales value - Pedicted sales value

By using R:

> Model

Call:
lm(formula = Sales ~ Population, data = data)

Coefficients:
(Intercept) Population  
102.029 2.124  

> Predicted_Sales = predict(Model,data)
> data.frame(Predicted_Sales)
Predicted_Sales
1 133.8878
2 148.7551
3 106.2770
4 129.6399
5 129.6399
6 119.0204
7 123.2682
8 142.3834
9 108.4009
10 153.0029
11 155.1268
12 123.2682
13 133.8878
14 148.7551
15 119.0204
16 138.1356
17 133.8878
18 150.8790
19 131.7638

> Ressidual = data.frame(data$Sales - Predicted_Sales)
> Ressidual
data.Sales...Predicted_Sales
1 14.112245
2 -14.755102
3 -52.276968
4 6.360058
5 12.360058
6 -10.020408
7 45.731778
8 -14.383382
9 -9.400875
10 -11.002915
11 -6.126822
12 18.731778
13 22.112245
14 -9.755102
15 -15.020408
16 -4.135569
17 32.112245
18 -23.879009
19 19.236152

If the residual sum is nearly equal to zero then the model is good

> sum(Residual)
[1] -4.405365e-13 nearly equal to zero

Residual Plot:

Graph line is a straight line means data follows the normality assumption.

>>>>>>>>>>> Best Luck >>>>>>>>>>>>


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