Many regions in North and South Carolina and Georgia have experienced rapid popu

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.

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.)

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

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

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

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.)

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.)

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?

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

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

Explanation / Answer

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.

Correlations

Food

Income

Income

.142

1

Size

.880

.166

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