STATISTICS FOR MANAGERS

Answers, Ch. 14

14.41

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

* * *

b. 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

* * *

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

* * *

d.

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, => 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

* * *

e.  

H₀: ß₁ = ß₂ = 0

H₁: At least one ß_j does not equal 0

Alpha = 0.05, n = 12

F _{k, n - k - 1} = (SSR / k) / (SSE / (n - k - 1))

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

F _{2, 12 - 2 - 1} = 28.53 > 4.26 => Reject H₀, accept H₁

* * *

f.

H₀: ß_j = 0

H₁: ß_j ≠ 0

Alpha = 0.05, n = 12

Alpha = 0.05 => Reject H₀ if t_{12 - 2 - 1} > 2.2622 or < -2.2622

ß₁:  t_{12 - 2 - 1} = 6.72 > 2.2622 => Reject H₀, accept H₁

ß₂:  t_{12 - 2 - 1} = 3.45 > 2.2622 => Reject H₀, accept H₁

* * *

g.

β_j ≈ b_j ± t_n-k-1 S_bj

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

ß₁ ~ 7.4 ± 2.2622 (1.1)

=> 4.9 < ß₁ < 9.9

ß₂ ~ 45 ± 2.2622 (13)

=> 15.5 < ß₂ < 74.5

* * *

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

* * *

j. r_adj² = 0.83

* * *

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.