Question

Fran’s Convenience Marts is located throughout the Erie, Pennsylvania metro area. Fran, the owner, wants to expand her businesses to other communities in northwest Pennsylvania and southeast New York, such as Jamestown, Corry, Meadville, and Warren. To prepare your presentation to the local bank, you would like to better understand the factors that make a particular discount store productive. Fran must do all the work on her own, so she won't be able to study all the discount stores. Therefore, he selects a random sample of 15 stores and records the average daily sales, the floor space (area), the number of parking spaces and the average income of the families in the region for each of the stores. The sample information is reported below.

Sampled Mart	Daily sales	Store area	Parking Spaces	Income
1	$1840	532	6	44
2	1746	478	4	51
3	1812	530	7	45
4	1806	508	7	46
5	1792	514	5	44
6	1825	556	6	46
7	1811	541	4	49
8	1803	513	6	52
9	1830	532	5	46
10	1827	537	5	46
11	1764	499	3	48
12	1825	510	8	47
13	1763	490	4	48
14	1846	516	8	45
15	1815	482	7	43

With the information above carry out the analysis required for the study that you must present to the bank regarding the best equation to estimate daily sales. Using all the information previously obtained by you:

a. indicate the correlation coefficients, identifying which is the best and the weakest among all the possible regressions and equations.
b. the regression errors obtained, identifying which is the best and the weakest among all the possible regressions and equations.
c. the required hypothesis tests
d.Present and identify which is the best equation to predict the monthly average purchase volume, explain why it is the best equation.
e.With the best estimated equation present the confidence interval to predict the monthly average purchase volume, when the Area of
the store is 585, the family income is 50,000 and the parking number is 10.

Answer 1

Answer:

1) The regression line is

y = b1x1 + c

Where y= Daily sales

b1=slope

x1=Store area

c=Intercept

In order to perform regression analysis using the data in Excel, follow the below steps

• Open Data Data Analysis Regression Analysis
• Select y variable range
• Select x1 variable range
• Select Output location
• Click OK

You will get an output as below

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.658651907
R Square	0.433822335
Adjusted R Square	0.390270207
Standard Error	22.8990747
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	1	5223.221	5223.221	9.960991	0.007582
Residual	13	6816.779	524.3676
Total	14	12040
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1363.024788	140.7961	9.68084	2.62E-07	1058.853	1667.196	1058.853	1667.196
X Variable 1	0.860639465	0.27269	3.156104	0.007582	0.271527	1.449751	0.271527	1.449751

As per above table Correlation Coefficient is R Square which is 43.4% indicates an week relationship.

Standers error is 22.89 and the fitted regression line is

y = 0.86x1 + 1363.02

2)

The 2nd regression line is

y = b1x1 + b2x2 + c

Where y= Daily sales

b1,b2=slope

x1=Store area

x2=Parking spaces

c=Intercept

In order to perform regression analysis using the data in Excel, follow the below steps

• Open Data Data Analysis Regression Analysis
• Select y variable range
• Select x1 and X2 variable ranges
• Select Output location
• Click OK

You will get an output as below

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.896657897
R Square	0.803995384
Adjusted R Square	0.771327948
Standard Error	14.02347906
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	2	9680.104	4840.052	24.61152	5.67E-05
Residual	12	2359.896	196.658
Total	14	12040
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1342.490177	86.3319	15.55034	2.57E-09	1154.389	1530.591	1154.389	1530.591
X Variable 1	0.772651585	0.168016	4.598669	0.000612	0.406575	1.138728	0.406575	1.138728
X Variable 2	11.63375746	2.443769	4.76058	0.000464	6.309242	16.95827	6.309242	16.95827

As per above table Correlation Coefficient is R Square which is 80.4% indicates a strong relationship.

Standers error is 14.02 and the fitted regression line is

y = 0.77x1 + 11.63x2 + 1342.49

3)

The 3rd regression line is

y = b1x1 + b2x2 +b3x3 + c

Where y= Daily sales

b1,b2,b3=slope

x1=Store area

x2=Parking spaces

x3=Family income

c=Intercept

In order to perform regression analysis using the data in Excel, follow the below steps

• Open Data Data Analysis Regression Analysis
• Select y variable range
• Select x1,x2 and X3 variable ranges
• Select Output location
• Click OK

You will get an output as below

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.913977941
R Square	0.835355677
Adjusted R Square	0.79045268
Standard Error	13.4242577
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	3	10057.68	3352.561	18.60356	0.000128518
Residual	11	1982.318	180.2107
Total	14	12040
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1480.744612	126.3042	11.72364	1.48E-07	1202.751032	1758.738	1202.751	1758.738
X Variable 1	0.731498876	0.16333	4.478642	0.000933	0.372010879	1.090987	0.372011	1.090987
X Variable 2	9.991487385	2.599961	3.842938	0.002733	4.26901275	15.71396	4.269013	15.71396
X Variable 3	-0.00230826	0.001595	-1.44748	0.175655	-0.005818119	0.001202	-0.00582	0.001202

As per above table Correlation Coefficient is R Square which is 83.5% which indicates a very strong relationship.

Standers error is 13.42 and the fitted regression line is

y = 0.73x1 + 9.99x2 -0.002x3 + 1480.74

The hypothesis test is

Null hypothesis : There is no relationship between the x variables

Alternate hypothesis : There is a relationship between the x variables

Hence considering above three regression lines third line indicates the highest Correlation Coefficient and the lowest standard error.However the p value for variable x3 is greater than 0.05 which is our significant level, hence we cannot take this variable into consideration.

So the final best fitted line is

y = 0.73x1 + 9.99x2 + 1480.74

so the monthly average sales for store 585 and parking number 10 is,

 y = 0.73x1 + 9.99x2 + 1480.74 

y= 0.73*585+9.99*10+1480.74

y=2007.69

Hence the monthly average purchase volume is $2008 .

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