Question

A national sporting good store wishes to use demographic information to predict its monthly sales, in $1000s. Thrity-eight, n=38, stores of the chain are randomly chosen across the country. It is known that each store is approximately the same size and carries the same merchandise.

The geographic area from which a store draws its customers is known as the customer base. One of the variables is the percentage of the customer base who have graduated from high school.

MonthlySalesi = β0 + β1PercentHSGradsi + ei
where,

MonthlySalesi - is the total sales in month i, in $1000s

PercentHSGradsi - is percentage of all customers in store i customer base that have graduated from high school

A least-squares regression was ran in R producing the following output:

Regression Analysis: MonthlySales versus PercentHSGrads

Predictor	Coef	SD Coef	T	P
Constant	-2970	1371
PercentHSGrads	59.66	17.67

S = 802.004 R-Sq =

Analysis of Variance
Source	DF	SS	MS	F	P
Regression		7333350
Residual Error		23155564
Total	37

(g) A store located at a local mall has recently discovered that 90% of its customer base has a high school diploma. With 95% confidence, estimate this store's monthly sales for the current month.
Note: You will need ∑38i=1PercentHSGradsi=2935.17 and ∑38i=1(PercentHSGradsi)^2=228777

Lower Bound:

Upper Bound:

The following answers that are already on Chegg have turned out to be incorrect for the above question: (2398.7, 2400.1) , (1871.6, 2927.2)

Answer 1

Ʃx = 2935.17

Ʃx² = 228777

n = 38

x̅ = Ʃx/n = 2935.17/38 = 77.2413

SSxx = Ʃx² - (Ʃx)²/n = 228777 - (2935.17)²/38 = 2060.6071

Sum of Square error, SSE = 23155564

Standard error, se = √(SSE/(n-2)) = √(23155564/(38-2)) = 802.0038

Predicted value of y at x = 90

ŷ = -2970 + (59.66) * 90 = 2399.4

Significance level, α = 0.05

Critical value, t_c = T.INV.2T(0.05, 36) = 2.0281

g) 95% Prediction interval :

Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 2399.4 - 2.0281*802.0038*√(1 + (1/38) + ((90 - 77.2413)²/(2060.6071))) = 689.4

Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 2399.4 + 2.0281*802.0038*√(1 + (1/38) + ((90 - 77.2413)²/(2060.6071))) = 4109.4