Question

Many regions 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.

Family		Food			Income			Size
1		$	4.24		$	73.98		2
2			4.08			54.90		2
3			5.76			59.43		4
4			3.48			52.02		1
5			4.20			65.70		2
6			4.80			53.64		4
7			4.32			79.74		3
8			5.04			68.58		4
9			6.12			165.60		5
10			3.24			64.80		1
11			4.80			138.42		3
12			3.24			125.82		1
13			6.37			77.58		7
14			5.51			170.74		7
15			6.60			126.42		6
16			5.40			141.30		3
17			6.00			36.90		5
18			5.40			56.88		4
19			3.36			71.82		1
20			4.68			69.48		3
21			4.32			54.36		2
22			5.52			87.66		5
23			4.56			38.16		3
24			5.40			43.74		7
25			7.10			37.46		6

Food and income are reported in thousands of dollars per year, and the variable size refers to the number of people in the household.

a-1. Develop a correlation matrix. (Round your answers to 3 decimal places. Negative amounts should be indicated by a minus sign.)

	Food Income Income Size

b-1. Determine the regression equation. (Round your answer to 3 decimal places.)

	The regression equation is: Food = + Income + size

b-2. How much does an additional family member add to the amount spent on food? (Round your answer to the nearest dollar amount.)

	Another member of the family adds to the food bill.

c-1. What is the value of R²? (Round your answer to 3 decimal places.)

	R2

c-2. Complete the ANOVA (Leave no cells blank - be certain to enter "0" wherever required. Round SS, MSto 4 decimal places and F to 2 decimal places.)

	Source DF SS MS F p-value Regression Error Total

c-3. 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.)

	H0 is rejected if F >

d-1. Complete the table given below. (Leave no cells blank - be certain to enter "0" wherever required. Round Coefficient, SE Coefficient, P to 4 decimal places and T to 2 decimal places.)c-4. Can we reject H₀: = ?₁ = ?₂ = 0?

	Predictor Coefficient SE Coefficient t p-value Constant Income Size

d-2. Would you consider deleting either of the independent variables?

	There is to delete a variable.

Answer 1

Part a-1

Required correlation matrix by using excel is given as below:

Correlations
	Food	Income
Income	.142	1
Size	.880	.166

Part b-1

Required regression equation is given as below:

Food = 3.207 – 0.000*Income + 0.479*Size

Regression output by using excel is given below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.880381811
R Square	0.775072133
Adjusted R Square	0.754624145
Standard Error	0.527012638
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	2	21.05540494	10.52770247	37.9045672	7.45549E-08
Residual	22	6.110331062	0.277742321
Total	24	27.165736
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.206622882	0.293228819	10.93556523	2.31525E-10	2.598503534	3.814742231
Income	-0.000126527	0.002731651	-0.046318845	0.963474011	-0.005791624	0.005538571
Size	0.479443914	0.055791163	8.5935458	1.76342E-08	0.363740124	0.595147703

Part b-2

Another member of the family adds $1 to the food bill.

(Coefficient of variable size is given as 0.4794, and it is rounded to nearest dollar as $1.)

Part c-1

The value of the coefficient of determination or R² is given as 0.775.

R² = 0.775

This means, about 77.5% of the variation in the dependent variable food is explained by the independent variables income and size.

Part c-2

Required ANOVA table is given as below:

Source	df	SS	MS	F	P-value
Regression	2	21.0554	10.5277	37.90	0.00
Error	22	6.1103	0.2777	0	0
Total	24	27.1657	0	0	0

(by using excel output)

Part c-3

Decision rule: We reject the null hypothesis H₀ if the P-value is less than the given level of significance or alpha value 0.05, otherwise we do not reject the null hypothesis.

F critical value = 3.443357 (by using F-table or excel)

Reject H₀ if F > 3.443357

We reject the null hypothesis H₀ if the test statistic F is greater than critical F value 3.443357.

Here, we get P-value = 0.00 < ? = 0.05, so we reject the null hypothesis

Part d-1

Required table of coefficients is given as below:

	Coefficients	Standard Error	t Stat	P-value
Intercept	3.2066	0.2932	10.94	0.0000
Income	-0.0001	0.0027	-0.05	0.9635
Size	0.4794	0.0558	8.59	0.0000

Part d-2

Yes, we would consider deleting the independent variable income, because this variable is not statistically significant as the p-value for this variable is given as 0.9635 which is greater than ? = 0.05.