Collins auto believes the number of cars sold in a day (Q) depends on two factors: the # of
hrs the dealership is open (H) and the # of salespersons working that day (S). After
collecting two months data (53 days), the manager estimates the following log-linear
model: Q= Q=aHbSc
a. Explain how to transform this log-linear model into a linear form that can be estimated
using multiple regression analysis.
b. How do you interpret coefficients b and c? If the dealership increases the # of
salespersons by 20%, what will be the % increase in daily sales?
c. Test the overall model for statistical significance at 5% significance level.
d. What % of the total variation in daily autosales is explained by this equation? what could
you suggest to increase this percentage?
e.Test intercept for statitical significance at the 5% level of significance. If H & S both = 0
are sales expected to be 0? Why/not?
f.If hrs of operation decreases 10% -the impact on sales is
Regression analysis chart follows:
Dependent Variable: LNQ R-Square F-Ration P-Value on P
Observations: 53 0.5452 29.97 0.0001
Variable Parameter Standard T-Ratio P-Valu
Estimate
Error
Intercept 0.9162
0.2413
3.80
0.0004
LNH
0.3517
0.1021
3.44
0.0012
LNS
0.2550
0.0785
3.25 .0021
Part A
𝑄 = 𝑎 𝐻𝑏 𝑆 𝑐
Taking natural logarithms on both sides
LN( 𝑄) = 𝐿𝑁( 𝑎) + 𝑏 LN( 𝐻) + 𝑐 LN( 𝑆)
That is
𝑄 ∗ = 𝑎∗ + 𝑏𝐻 ∗ + 𝑐𝑆 ∗
where, 𝑄 ∗ = 𝐿𝑁( 𝑄), 𝑎∗ = 𝐿𝑁( 𝑎) , 𝐻 ∗ = Ln(𝐻) 𝑎𝑛𝑑 𝑆 ∗ = LN( 𝑆)
Now we have transformed the log linear model into a linear model.
Part B
𝜕𝐿𝑁( 𝑄)
𝑏 = 𝜕 𝐿𝑁(𝐻) is the rate of proportional change 𝑄 with respect to proportional change in 𝐻.
For every 1% increase in 𝐻, the sales 𝑄 increases by b%.
𝜕𝐿𝑁( 𝑄)
𝑐 = 𝜕 𝐿𝑁( 𝑆) is the rate of proportional change 𝑄 with respect to proportional change in S.
For every 1% increase in S, the sales 𝑄 increases by c%.
If the number of sale persons increases by 20%, then the sales increases by 20 c%.
Part C
Testing for statistical significance of the model
Null hypothesis 𝐻0 : 𝑡ℎ𝑒 𝑚𝑜𝑑𝑒𝑙 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑝𝑙𝑎𝑖𝑛 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒.
Alternative hypothesis: 𝐻1 : the null hypothesis is false.
𝐹 = 29.97 and 𝑝 − 𝑣𝑎𝑙𝑢𝑒 = 0.0001 < 0.05
F is significant at the 5% level.
So reject the null hypothesis.
The model is statistically significant.
Part D
54.52% of the total variation in auto sales is explained by this equation. If we include the other
explanatory variables such as advertising expenditure, price of fuel etc in to the model the percentage of
variation explained by the model can be increased.
Part E
To test the statistical significance of the intercept, the t-value is 3.80 with p-value = 0.0004 < 0.05.
So t is significant.
So the intercept is statistically significant.
If both 𝐻 and 𝑆 are equal to 0, then the estimated sales is
𝑄 = 𝑎 𝐻 𝑏 𝑆 𝑐 = (0.9162) × (0)0.3517 × (0)0.2550 = 0
Part F
If hours of operation decreases by 10%, then sales decreases by 10 𝑏% = 10 × 0.3517% = 3.517%