STATISTICS FOR MANAGERS

Answers, Chapter 14

14.41

The marketing department of a large supermarket chain faced the business problem of determining the effect on the sales of pet food of shelf space and whether the product was placed at the front (=1) or back (=0) of the aisle.  Data are collected from a random sample of 12 equal-sized stores.  The results are in the following table:

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a. State the multiple regression equation that predicts sales based on shelf space and location

 Sales_i = ß₀ + ß₁ Space_i + ß₂ Location_i + €_i

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b. Interpret the regression coefficients in (a).

 Sales_i = 130 + 7.4 Space_i + 45 Location_i

b₁ = 7.4 => Sales will increase by $7.40 for every 1 foot increase in shelf space, holding the location constant

b₂ = 45 => Sales will increase by $45 if the display is at the front of the aisle as opposed to the back, holding the amount of space constant

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c. Predict the weekly sales of pet food for a store with 8 feet of shelf space situated at the back of the aisle.

Sales = 130 + 7.4 (8) + 45 (0) = $189.20

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d.  Perform a residual analysis on the results and determine whether the regression assumptions are valid.

Heteroscedasticity

Neither of the residual plots (for space or for place (location)) shows the fan shape that would be indicative of heteroscedasticity

Autocorrelation

n = 12, k = 2, α = 0.05 => d_L = 0.95, d_U = 1.54

d = 2.38 => No autocorrelation

Multicollinearity

VIF < 5 => No multicollinearity

5 < VIF < 10 => Uncertain

VIF > 10 => Multicollinearity

VIF = 1.00 => No multicollinearity

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e.  Is there a significant relationship between sales and the two independent variables (shelf space and aisle position) at the 0.05 level of significance?

H₀: ß₁ = ß₂ = 0

H₁: At least one ß_j does not equal 0

α = 0.05, n = 12

F_STAT = (SSR / k) / (SSE / (n - k - 1))

α = 0.05 => F _{2, 12 - 2 - 1, 0.05} = 4.26 => Reject H₀ if F_STAT > 4.26

F_STAT = 28.53 > 4.26 => Reject H₀, accept H₁

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f. At the 0.05 level of significance, determine whether each independent variable  makes a contribution to the regression model.  

H₀: ß_j = 0

H₁: ß_j ≠ 0

α = 0.05, n = 12

α = 0.05 => t_{12 - 2 - 1, 0.05/2} = 2.2622 =>  Reject H₀ if t_STAT > 2.2622 or < -2.2622

ß₁:  t_STAT = 6.72 > 2.2622 => Reject H₀, accept H₁

ß₂:  t_STAT = 3.45 > 2.2622 => Reject H₀, accept H₁

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g.  Construct and interpret 95% confidence interval estimates of the population slope for the relationship between sales and shelf space and between sales and aisle location.

β_j ≈ b_j ± t_n-k-1,α/2 S_bj

95% C.I. => t_{12 - 2 - 1, 0.05/2} = 2.2622

ß₁ ~ 7.4 ± 2.2622 (1.1)

=> 4.9 < ß₁ < 9.9

ß₂ ~ 45 ± 2.2622 (13)

=> 15.5 < ß₂ < 74.5

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i. Compute and interpret the meaning of the coefficient of multiple determination, r².

r² = 0.86 => 86% of the variation in sales is explained by the multiple regression model with shelf space and location as independent variables

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j.  Calculate the adjusted r².

r_adj² = 0.83

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m.  The slope of shelf space with sales is the same regardless of whether the display is in the front or the back of the store.