The following table shows the estimated equation of demand for ugali in restaurants around mzumbe area based on 69 observation.Figure in brackets are t-ratios
Qu=4.34
(2.91)
-0.37lnPu+0.27lnPr+0.18lnM
(3.78) (1.59) (4.81)
Where
Pu =price of ugali per plate;Pr=Price of rice per plate; M=per capita income
Given ESS=19650.5 and TSS=43979.1
Required.
a) determine which variables are significantly different from zero at the 5% level and interpret their coefficients
b) compute and interpret R-squared.
To determine which variables are significantly different from zero at the 5% level and interpret their coefficients, we can look at the t-ratios provided in the parentheses.
a) Significance of Variables:
In statistical analysis, the t-statistic measures the significance of each variable. A t-ratio above 2 (in absolute value) indicates that the variable is significantly different from zero at the 5% level of significance.
Given the t-ratios in the parentheses for each variable:
- The t-ratio for lnPu is 2.91, indicating that the coefficient for lnPu is significantly different from zero at the 5% level.
- The t-ratio for lnPr is 1.59, indicating that the coefficient for lnPr is not significantly different from zero at the 5% level. In other words, the variable lnPr does not have a statistically significant impact on the demand for ugali in restaurants.
- The t-ratio for lnM is 4.81, indicating that the coefficient for lnM is significantly different from zero at the 5% level.
Therefore, Pu and M are the variables that are significantly different from zero at the 5% level.
Interpretation of Coefficients:
- The coefficient for lnPu (-0.37) suggests that as the price of ugali per plate (Pu) increases by 1%, the demand for ugali in restaurants decreases by approximately 0.37%, holding other variables constant.
- The coefficient for lnM (0.18) suggests that as the per capita income (M) increases by 1%, the demand for ugali in restaurants increases by approximately 0.18%, holding other variables constant.
b) R-squared:
R-squared (R^2) is a statistical measure that represents the proportion of variance in the dependent variable that is predictable from the independent variables. It gives an indication of how well the estimated equation fits the observed data.
To compute R-squared, we need to calculate the explained sum of squares (ESS) and the total sum of squares (TSS) using the given values.
ESS (Explained Sum of Squares) = 19650.5
TSS (Total Sum of Squares) = 43979.1
R-squared can be calculated as the ratio of ESS to TSS:
R^2 = ESS / TSS
Using the provided values:
R^2 = 19650.5 / 43979.1 ≈ 0.4464
Interpretation of R-squared:
R-squared represents the proportion of the variance in the demand for ugali in restaurants that can be explained by the independent variables included in the estimated equation. In this case, approximately 44.64% of the variance in the demand for ugali can be explained by the variables lnPu, lnPr, and lnM. The remaining 55.36% of the variance is attributed to other factors or sources of variation not captured in the equation.