Question

In: Statistics and Probability

Fran’s Convenience Marts is located throughout the Erie, Pennsylvania metro area. Fran, the owner, wants to...

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(Thousands of Dollars)

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.

Only c and e (Important)

Solutions

Expert Solution

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

c) to e)

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.

e)

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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