STATISTICS FOR MANAGERS

Answers, Chapter 14

14.45

Zagat's publishes restaurant ratings for various locations in the United States.  The data has the Zagat rating for food, decor, and service as well as the cost per person for a sample of 50 restaurants located in a city and 50 restaurants located in a suburb.  Develop a regression model to predict the cost per person based on the ratings and a dummy variable for whether the restaurant is in a city (= 0) or in a surburb (= 1).

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a. State the multiple regression equation that predicts cost based on location and the ratings of food, decor, and service.

 Cost_i = ß₀ + ß₁ Location_i + ß₂ Food_i + ß₃ Decor_i + ß₄ Service_i + €_i

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b. Interpret the regression coefficients in (a).

 Cost_i = -31.83 - 6.33 Location_i + 0.20 Food_i + 1.97 Decor_i + 1.89 Service_i

b₁ = -6.33 => Cost will decrease by $6.33 if the restaurant is in the suburbs, holding all other independent variables constant

b₂ = 0.20 => Cost will increase by $0.20 for a 1-unit increase in the food rating, holding all other independent variables constant

b₃ = 1.97 => Cost will increase by $1.97 for a 1-unit increase in the decor rating, holding all other independent variables constant

b₄ = 1.89 => Cost will increase by $1.89 for a 1-unit increase in the service rating, holding all other independent variables constant

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c. Predict the cost for a restaurant with a food rating of 18, a decor rating of 20, and a service rating of 22 that is located in a city.

Cost_i = -31.83 - 6.33 (0) + 0.20 (18) + 1.97 (20) + 1.89 (22)  = $52.67

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

Heteroscedasticity

None of the residual plots shows the fan shape that would be indicative of heteroscedasticity

.

Autocorrelation

n = 100, k = 4, α = 0.05 => d_L = 1.59, d_U = 1.76

d = 1.66 => Uncertain

.

Multicollinearity

VIF < 5 => No multicollinearity

5 < VIF < 10 => Uncertain

VIF > 10 => Multicollinearity

VIF_Location = 1.09 => No multicollinearity

VIF_Food = 2.07 => No multicollinearity

VIF_Decor = 1.50 => No multicollinearity

VIF_Service = 2.56 => No multicollinearity

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e.  Is there a significant relationship between cost and the independent variables at the 0.05 level of significance?

H₀: ß₁ = ß₂ = ß₂ = ß₄ =0

H₁: At least one ß_j does not equal 0

α = 0.05, n = 100

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

α = 0.05 => F _{4, 100 - 4 - 1, 0.05} = 2.45 => Reject H₀ if F_STAT > 2.45

F_STAT = 38.67 > 2.45 => 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_{100 - 4 - 1, 0.05/2} = 1.9853 =>  Reject H₀ if t_STAT > 1.9853 or < -1.9853

ß₁:  t_STAT = -3.58 < -1.9853 => Reject H₀, accept H₁

ß₂:  t_STAT = 0.35 not > 1.9853  => Do not reject H₀

 ß₃:  t_STAT = 6.82 > 1.9853  => Reject H₀, accept H₁

ß₂:  t_STAT = 3.04 > 1.9853  => Reject H₀, accept H₁

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g.  Construct and interpret 95% confidence interval estimates of the population slope for the relationship between cost and all the independent variables.

β_j ≈ b_j ± t_n-k-1,α/2 S_bj

95% C.I. => t_{100 - 4 - 1, 0.05/2} = 1.9853

ß₁ ~ -6.33 ± 1.9853 (1.76)

=> -9.83 < ß₁ < -2.82

ß₂ ~ 0.20 ± 1.9853 (0.59)

=> -0.96 < ß₂ < 1.37

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

r² = 0.62 => 62% of the variation in cost is explained by the multiple regression model with location, food rating, decor rating, service rating

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j.  Calculate the adjusted r².

r_adj² = 0.60

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m. What assumption about the slope of cost with each of the rating independent variables do you need to make in this problem?.

The slope of cost with each of the rating independent variables is the same regardless of whether the restaurant is in a city or is in the suburbs.