Question

1. The marketing manager of a large supermarket chain would like to use shelf space to predict the sales of a specialty pet food. Data are collected from a random sample of 8 equal-sized stores, with the following results:

Store	Shelf Space (in square feet)	Weekly Sales (in Dollars)
1	4	120
2	4	150
3	8	160
4	8	180
5	12	200
6	16	210
7	16	240
8	20	260

Use Excel to find the regression results for this problem. Include Excel results with your submission.

a. at the 0.05 level of significance, is there evidence of a linear relationship between shelf space and weekly sales?

b. construct a 95% confidence interval estimate of the population slope, β₁.

Answer 1

Data Analysis > Data > Regression

Output using excel:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.96199242
R Square	0.92542941
Adjusted R Square	0.91300098
Standard Error	13.8346612
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	1	14251.6129	14251.6129	74.4606742	0.00013338
Residual	6	1148.3871	191.397849
Total	7	15400
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	106.612903	10.8308984	9.84340353	6.337E-05	80.1106497	133.115157
X	7.58064516	0.87850186	8.62905987	0.00013338	5.43102854	9.73026178

Slope, b1 = 7.5806452

Standard error of slope, se(b1) = 0.87850186

a) Null and alternative hypothesis:

Ho: β₁ = 0 ; Ha: β₁ ≠ 0

Test statistic:

t = b1/se(b1) = 7.5806/0.8785 = 8.6291

p-value = T.DIST.2T(ABS(8.6291), 6) = 0.0001

Conclusion:

p-value < α Reject the null hypothesis.

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Q2: Significance level, α = 0.05

Critical value, t_c = T.INV.2T(0.05, 6) = 2.4469

95% Confidence interval for slope:

Lower limit = b1 - tc*se(b1) = 7.5806 - 2.4469*0.8785 = 5.4310

Upper limit = b1 + tc*se(b1) = 7.5806 + 2.4469*0.8785 = 9.7303