Question

 

The manager of a discount store would like to determine whether there is a relationship between the number of customers who visit the store each day and the dollar value of sales that day. A random sample of n = 20 days was taken and the number of customers in the store and the dollar value of sales were recorded for each day. (Note: n = the sample size = 20). The sample results are shown below.

Day	Number of Customers	$Sales	Day	Number of Customers	$Sales
1	907	11,200	11	679	7,630
2	926	11,050	12	872	9,430
3	506	6,840	13	924	9,460
4	741	9,210	14	607	7,640
5	789	9,420	15	452	6,920
6	889	10,080	16	729	8,950
7	874	9,450	17	794	9,330
8	510	6,730	18	844	10,230
9	529	7,240	19	1010	11,770
10	420	6,120	20	621	7,410

As a part of the study, the manager would like to estimate the correlation between the two variables, number of customers and $Sales. She would also like to conduct a hypothesis test to determine if the linear relationship between $Sales and the number of customers is positive.   

7. The calculated sample correlation coefficient, r, for the two variables $Sales and number of customers = 0.955. The correct specification for the null and alternative hypothesis to determine if the true population correlation is positive would be:

A. Ho: r = 0 vs. Ha: r ≠ 0

B. Ho: r < 0 vs. Ha: r > 0

C. Ho: ρ = 0 vs. Ha: ρ ≠ 0

D. Ho: ρ < 0 vs. Ha: ρ > 0

E. Ho: ρ > 0 vs. Ha: ρ < 0

8. The critical value for the test statistic for the test of positive correlation between the two variables with n = 20 and α = 0.05 is

a. t = 1.7247

b. t = -1.7247

c. t = 1.7341

d. t = -1.7341

e. t = 2.1009

The calculated value of the test statistic computed from the sample correlation coefficient of r = 0.955 is

a. 5.79                        b. 13.66                       c. 18.9             d. 33.23

The simple linear regression was developed with the number of customers as the independent variable and the $Sales as the dependent variable. The following results were obtained:

	Coefficients	Standard Error	t Stat
Intercept	2423	480.96
Customers	8.7	0.64

Use the information provided to develop a 90% confidence interval estimate for the true value of the slope coefficient, β1

A. 7.6 < β1 < 9.8

B. 7.4 < β1 < 10.1

C. 14121.6 < β1 < 3433.5

D. 8.7 < β1 < 9.34

E. 7.6 < β1 < 8.24

The simple linear regression of the data on $Sales and number of customers produced the following ANOVA output:

ANOVA
Source	df	SS	MS	F	Significance F
Regression		46833541
Residual
Total		51360495

 

Complete as much of the ANOVA table as you need to answer the following two questions (Questions 11 and 12):

What percent of the variation in the dependent variable $ Sales is explained by the regression model?

A. 100%

B. 91.2%

C. 95.5%

D. 90%

The calculated value of the test statistic used to test the following null and alternative hypotheses

Ho: The overall regression model is not significant

Ha: The overall regression model is significant

would be

A. F = 186.22

B. F = 4.4139

C. F = 2.1009

D. Z = 1.645

E. Z = 2.288

Use the following information to answer the next 2 questions (Questions 13 and 14)

A simple linear regression was developed with the number of customers as the independent variable and the $Sales as the dependent variable. The following results were obtained:

	Coefficients	Standard Error	t Stat
Intercept	2423	480.96
Customers	8.7	0.64

For a week having 750 customers the point estimate for the $ value of Sales predicted by the simple linear regression equation would be (that is, predict $Sales for Customers = 750)

A. $750

B. $6525

C. $2332

D. $8948

Compute the 95% confidence interval for the expected (average) sales for many days where the number of customers averages 700 (that is, Xp = 700). In addition to the information provided above, the following is also available to assist you in constructing the confidence interval:

x = 700.    

E(x-x)² = 614602. This is the DEVSQ formula result from Excel.

You will need to use the partially provided output earlier to find the other necessary inputs. Please show your work: The 95% confidence interval for the expected (average) sales for an average of many days where there are 700 customers is:

Answer 1

7)

C. Ho: ρ = 0 vs. Ha: ρ ≠ 0

8)
t ==T.INV.2T(0.05,18)
= 2.1009
option E)

from Excel regression result

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9549132
R Square	0.91185922
Adjusted R Square	0.90696251
Standard Error	501.4952145
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	1	46833540.9	46833540.9	186.2187504	6.20621E-11
Residual	18	4526954.104	251497.4502
Total	19	51360495
	Coefficients	Standard Error	t Stat	P-value	Lower 90.0%	Upper 90.0%
Intercept	2423.044396	480.9646094	5.037885009	8.55388E-05	1589.021171	3257.067621
Number of Customers	8.729338171	0.639690078	13.64619912	6.20621E-11	7.620074889	9.838601454

TS = 13.66
option B) is correct

90% confidence interval for b1
7.620074889   9.838601454
A. 7.6 < β1 < 9.8 is correct

ANOVA
	df	SS	MS	F	Significance F
Regression	1	46833540.9	46833540.9	186.2187504	6.20621E-11
Residual	18	4526954.104	251497.4502
Total	19	51360495

What percent of the variation in the dependent variable $ Sales is explained by the regression model?

= R^2 = 0.9119

B) 91.2 %

The calculated value of the test statistic used to test the following null and alternative hypotheses

F = 186.2188

A. F = 186.22   is correct

	Coefficients	Standard Error	t Stat
Intercept	2423.044396	480.9646094	5.037885009
Number of Customers	8.729338171	0.639690078	13.64619912