Lenny's, a national restaurant chain, conducted a study of the factors affecting demand (sales). The following variables were defined and measured for a random sample of 30 of its restaurants:
Y= Annual restaurant sales ($000)
X₁= Disposable personal income (per capita) of residents within 5 mile radius
X₂= License to sell beer/wine (0 = No, 1 = Yes)
X₃= Location (within one-half mile of interstate highway-0 = No, 1 = Yes)
X₄= Population (within 5 mile radius)
X₅= Number of competing restaurants within 2 mile radius
The data were entered into a computerized regression program and the following results were obtained: Questions:
(a)Give the regression equation for predicting restaurant sales.
(b)Give an interpretation of each of the estimated regression coefficients.
(c)Which of the independent variables (if any) are statistically significant at the .05 level in "explaining"restaurant sales?
(d)What proportion of the variation in restaurant sales is "explained"by the regression equation?
(e)Perform an F-test (at the .05 significance level) of the overall explanatory power of the regression model.

Question

Lenny's, a national restaurant chain, conducted a study of the factors affecting demand (sales). The following variables were defined and measured for a random sample of 30 of its restaurants:
Y= Annual restaurant sales ($000)
X₁= Disposable personal income (per capita) of residents within 5 mile radius
X₂= License to sell beer/wine (0 = No, 1 = Yes)
X₃= Location (within one-half mile of interstate highway-0 = No, 1 = Yes)
X₄= Population (within 5 mile radius)
X₅= Number of competing restaurants within 2 mile radius
The data were entered into a computerized regression program and the following results were obtained:

Questions:
(a)Give the regression equation for predicting restaurant sales.
(b)Give an interpretation of each of the estimated regression coefficients.
(c)Which of the independent variables (if any) are statistically significant at the .05 level in "explaining"restaurant sales?
(d)What proportion of the variation in restaurant sales is "explained"by the regression equation?
(e)Perform an F-test (at the .05 significance level) of the overall explanatory power of the regression model.

Quizplus · Accepted Answer

The Answer is (a)Y = .363 + .00275X₁ + 76.65X₂ + 164.3X₃ + .00331X₄ -46.2X₅
(b)a = .363,Value of dependent variable (Y) when all independent variables (X's) are equal to zero.
b₁ = .00275,For a one dollar increase in per capita disposable income, expected restaurant sales will increase by .00275(* $1000) = $2.7
b₂ = 76.65 ,Expected annual restaurant sales are 76.65(* $1000) = $76,650 higher for a restaurant with a license to sell beer/wine than for one without such a license.
b₃ = 164.3 ,Expected annual restaurant sales are 164.3(* $1000) = $164,300 higher for a restaurant located within one-half mile of an interstate highway.
b₄ = .00331 ,For a one person increase in population, expected restaurant sales will increase by .00331(* $1000) = $3.31.
b₅ = -46.2 ,For a one unit increase in the number of restaurants within a 2-mile radius, expected annual restaurant sales decrease by 46.2(* $1000) = $46,200.
(c)H₀:

\beta

_i = 0
H₁:

\beta

_i

eq

0
Reject H₀ if t > t_{.025, 24} = 2.064 or t <-2.064.
The t-values of X₁ and X₄ are greater than +2.064 (and the t-value of X₅ is less than -2.064). Therefore X₁, X₄ , and X₅ are statistically significant at the .05 level in "explaining"restaurant sales.
(d)According to the R-SQUARE statistic, 79 percent of the variation in restaurant sales is "explained"by the regression equation.
(e)H₀: All b_i = 0 (no relationship)
H₁: At least one b_i

eq

0
Reject H₀ if F > F_{.05, 5, 24} = 2.62
The F-STAT is equal to 18.18, which exceeds 2.62. Therefore, one rejects the null hypothesis (at the .05 significance level) and concludes that the five independent variables "explain"a significant proportion of the variation in restaurant sales.

Lenny's, a National Restaurant Chain, Conducted a Study of the Factors