Exercise 13-26 H 0 . Many urban regions have experienced rapid population growth
ID: 3357940 • Letter: E
Question
Exercise 13-26
H0.
Many urban regions have experienced rapid population growth over the last 10 years. It is expected that growth will continue over the next 10 years. This has resulted in many of the large grocery store chains building new stores. The Kelley’s Super Grocery Stores Inc. chain is no exception. The director of planning for Kelley’s Super Grocery Stores wants to study adding more stores. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family. The director gathered the following sample information:Explanation / Answer
Solution:
Here, we have to use regression analysis for the given scenario. The regression model (using excel) for estimation of the dependent variable food based on the independent variables income and size is given as below:
Regression Statistics
Multiple R
0.899395319
R Square
0.808911941
Adjusted R Square
0.791540299
Standard Error
0.418233298
Observations
25
ANOVA
df
SS
MS
F
P-value
Regression
2
16.29024399
8.145122
46.56508
1.24042E-08
Residual
22
3.848220007
0.174919
Total
24
20.138464
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
2.666164411
0.290259182
9.185461
5.53E-09
2.064203713
3.268125109
Income ($)
0.006554276
0.002634135
2.488208
0.020898
0.001091413
0.012017138
Size
0.336743459
0.0398169
8.4573
2.32E-08
0.254168262
0.419318656
Correlation matrix is given as below:
Food ($)
Income ($)
Size
Food ($)
1
Income ($)
0.433187
1
Size
0.868986
0.239399
1
Questions:
Part a.1
The required correlation matrix is given as below:
Food ($)
Income ($)
Income ($)
0.433
1.000
Size
0.869
0.239
Part a.2
The correlation coefficient between independent variables size and income is given as 0.239, which implies a low weak positive correlation exists between independent variables. So, there is a very low multicollinearity exists.
Part b.1
Regression equation is given as below:
Food = 2.666 + 0.007*Income + 0.337*Size
(By using regression output given above)
Part b.2
For the given regression equation, y-intercept is given as 2.666 which implies the value of food when there is no any income and size. The slope for the variable income is given as 0.007 which indicate the increase in food price as per one dollar increase in income. The slope for the independent variable size is given as 0.337 which indicate the average increase in food price as per increment of one person in family size.
Part b.3
Another member of the family adds $0.337 to the food bill.
Part c.1
The value of R square or coefficient of determination is given as below:
R2 = 0.809
Part c.2
H0 is rejected if F>3.44
(by using F-table or excel with = 0.05, df1 = 2, df2 = 22)
Part c.3
Test statistic is given as below:
F = 46.57
(By using F = MSR/MSE = 8.145122/0.174919 = 46.56508)
Part c.4
F calculated = 46.57 > F critical = 3.44
We reject the null hypothesis H0 because test statistic value F is greater than critical F value.
Part d
No, we would not consider deleting either of the independent variables because both variables are statistically significant at 5% level of significance.
Regression Statistics
Multiple R
0.899395319
R Square
0.808911941
Adjusted R Square
0.791540299
Standard Error
0.418233298
Observations
25
ANOVA
df
SS
MS
F
P-value
Regression
2
16.29024399
8.145122
46.56508
1.24042E-08
Residual
22
3.848220007
0.174919
Total
24
20.138464
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
2.666164411
0.290259182
9.185461
5.53E-09
2.064203713
3.268125109
Income ($)
0.006554276
0.002634135
2.488208
0.020898
0.001091413
0.012017138
Size
0.336743459
0.0398169
8.4573
2.32E-08
0.254168262
0.419318656
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