Many regions along the coast in North and South Carolina and Georgia have experi
ID: 3360675 • Letter: M
Question
Many regions along the coast in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has motivated many of the large grocery store chains to build new stores in the region. 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 in this region. 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.
Food and income are reported in thousands of dollars per year, and the variable size refers to the number of people in the household.
Develop a correlation matrix. (Round your answers to 3 decimal places. Negative amounts should be indicated by a minus sign.)
How much does an additional family member add to the amount spent on food? (Round your answer to the nearest dollar amount.)
State the decision rule for 0.05 significance level. H0: = 1 = 2 = 0; H1: Not all i's = 0. (Round your answer to 2 decimal places.)
Complete the given below table. (Leave no cells blank - be certain to enter "0" wherever required. Round Coef, SE Coef, P to 3 decimal places and T to 2 decimal places.)
Many regions along the coast in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has motivated many of the large grocery store chains to build new stores in the region. 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 in this region. 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
Question a.1
The required correlation matrix is given as below:
Food
Income
Income
0.315318
1
Size
0.69226
0.055012
(by using excel)
Question a.2
The correlation coefficient between the two independent variables size and income is given as 0.055, which is very low and therefore there is no any problem with multicollinearity. Multicollinearity is exists if there is a significant relationship exists between the given independent variables.
Question b.1.
Regression output by using excel for the given scenario is summarized as below:
Regression Statistics
Multiple R
0.745866798
R Square
0.55631728
Adjusted R Square
0.515982487
Standard Error
0.669387192
Observations
25
ANOVA
df
SS
MS
F
Significance F
Regression
2
12.36025732
6.18012866
13.7924913
0.000131159
Residual
22
9.857742681
0.44807921
Total
24
22.218
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
3.096421403
0.399310429
7.75442156
9.8665E-08
2.268302261
3.924540544
Income
0.007090153
0.003626384
1.95515818
0.06338593
-0.000430506
0.014610813
Size
0.340154377
0.071465133
4.75972496
9.4416E-05
0.191944763
0.488363991
The regression equation is given as below:
Food = 3.096 + 0.007*Income + 0.340*Size
Question b.2
About $0.34 add to the amount spent on food as per an additional family member added.
(See, coefficient of family size is given as 0.340.)
Part c.1
The value of R^2 or coefficient of determination is given as below:
R2 = 0.746
About 74.6% of the variation in the dependent variable food expenditure is explained by the independent variables income of the person and family size.
Part c.2
Decision rule is given as below:
DF1 = 2, DF2 = 22, = 0.05
Critical value = 3.443356779
(By using F-table)
H0 is rejected if F > 3.44
Part c.3
Required ANOVA table is given as below:
ANOVA
df
SS
MS
F
P-value
Regression
2
12.36
6.18
13.79
0.000
Residual
22
9.86
0.45
Total
24
22.22
Part c.4
P-value is less than = 0.05
So, we reject the null hypothesis H0
Part d.1
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
3.096421403
0.399310429
7.75442156
9.8665E-08
2.268302261
3.924540544
Income
0.007090153
0.003626384
1.95515818
0.06338593
-0.000430506
0.014610813
Size
0.340154377
0.071465133
4.75972496
9.4416E-05
0.191944763
0.488363991
Food
Income
Income
0.315318
1
Size
0.69226
0.055012
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