Chez Henri is a restaurant chain that operates in 40 different cities. It hired

Question

Chez Henri is a restaurant chain that operates in 40 different cities. It hired an economist to estimate the factors affecting the demand for its sales. The following equation was estimated using cross sectional data from each of its 40 restaurants.

Y
Annual restaurant sales (in thousands)

X1
Disposable per capital income (in thousands) of the residents living within 5 miles of a restaurant

X2
Population (in thousands) within a 5-mile radius of a restaurant

X3
Number of competing restaurants within a 5-mile radius


The following information was obtained from the regression analysis:
Multiple R: 0.92
R-Square: 0.85
Std. Error of Est.: 0.40

Analysis of Variance



DF
Sum Squares
Mean Sqr.
F-Stat

Regression
3
220
73.3
18.2

Residual
36
60
1.7





Variable
Coefficient
Std. Error
T-Value


Constant
0.4
0.2
2.0


X1
0.01
0.004
2.5


X2
0.02
0.015
1.3


X3
-20.2
4.50
-4.6



Answer the following questions:
a. Give the estimated demand equation for predicting restaurant sales.
b. Provide an interpretation for each of the regression coefficients.
c. Which of the coefficients are statistically significant and which are not? Explain.
d. What percent of variation are restaurant sales explained by this equation?

Explanation / Answer

(a) Estimated demand equation:


Y = 0.4 + 0.01X 1 + 0.02 X 2 20.2 X 3

(b) Let us consider each coefficient one by one:
When X1, i.e., disposable per capita income of the residents living within 5 miles of a
restaurant increases by one unit, sales of the restaurant namely, Y increases by 0.01 units.
When X2, i.e., population within 5 mile radius of a restaurant increases by one unit, sales
of the restaurant namely, Y increases by 0.02 units.
When X3, i.e., number of competing restaurants within a 5 mile radius increases by one
unit, sales of the restaurant namely, Y decreases by 20.2 units.

(c) In order to comment on the significance of a coefficient, we compare its estimated t-

statistic with the tabular t-statistic at 1% and 5% level of significance corresponding to (n
k) degrees of freedom. Here, n = 40 and k = 4. Therefore, degrees of freedom = 36.
Now, t0.005,36 = 2.7 and t0.025,36 = 2.02
Estimated t-statistic of X1 = 2.5

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