Answer :
Sure, let's go step-by-step through each part:
### a. Interpret the estimated logit model.
The estimated logit model is given by:
[tex]\[ \hat{L}_i = -2.77231 + 0.347582 \cdot \ln(\text{Income}) \][/tex]
Where:
- [tex]\(\hat{L}_i\)[/tex] is the estimated logit which is a linear combination of the coefficients.
- The coefficient for the intercept is -2.77231.
- The coefficient for the logarithm of income is 0.347582.
Interpretation:
- The intercept [tex]\(-2.77231\)[/tex] represents the log-odds of car ownership when the logarithm of income is zero [tex]\( \ln(\text{Income}) = 0 \)[/tex]. Since [tex]\(\ln(1) = 0\)[/tex], this essentially means the log-odds of car ownership when income is 1 unit of currency (assuming the unit used allows logarithm to make sense).
- The coefficient of [tex]\(\ln(\text{Income})\)[/tex], which is 0.347582, indicates how the log-odds of car ownership change with respect to a one-unit change in the logarithm of income.
- The t-values (in parentheses under the coefficients) indicate the statistical significance of these coefficients. The t-value for the intercept is -3.35, and for the slope (log of income) it is 4.05.
### b. From the estimated logit model, how would you obtain the expression for the probability of car ownership?
To convert the logit function into a probability, we use the logistic function. The probability [tex]\( \hat{P}_i \)[/tex] of car ownership can be obtained by transforming the logit [tex]\(\hat{L}_i\)[/tex]:
[tex]\[ \hat{P}_i = \frac{1}{1 + e^{-\hat{L}_i}} \][/tex]
Substituting [tex]\( \hat{L}_i \)[/tex] in, we get:
[tex]\[ \hat{P}_i = \frac{1}{1 + e^{-(-2.77231 + 0.347582 \cdot \ln(\text{Income}))}} \][/tex]
### c. What is the probability that a household with an income of 20,000 will own a car? And at an income level of 25,000? What is the rate of change of probability at the income level of 20,000?
Given the calculated values:
- For income of 20,000:
[tex]\[ \ln(\text{Income})_{20k} = 9.903487552536127 \][/tex]
[tex]\[ \text{logit}_{20k} = 0.6699640104856122 \][/tex]
[tex]\[ \text{probability}_{20k} = 0.6614951005017523 \][/tex]
- For income of 25,000:
[tex]\[ \ln(\text{Income})_{25k} = 10.126631103850338 \][/tex]
[tex]\[ \text{logit}_{25k} = 0.7475246923385082 \][/tex]
[tex]\[ \text{probability}_{25k} = 0.678639102980446 \][/tex]
- The rate of change of probability at the income level of 20,000 is:
[tex]\[ \text{rate of change}_{20k} = 0.07783032943385645 \][/tex]
### d. Comment on the statistical significance of the estimated logit model.
The model's statistical significance can be inferred from:
- The t-values for the intercept [tex]\(-2.77231\)[/tex] and the slope [tex]\(0.347582\)[/tex] are -3.35 and 4.05, respectively. A high t-value (in absolute terms) indicates that the coefficient is significantly different from zero. Given that the critical t-value for common significance levels (e.g., 1.96 for 5% significance) is much lower than the given t-values, both the intercept and slope coefficients are statistically significant.
- The chi-square ([tex]\( \chi^2 \)[/tex]) value for the goodness of fit is 16.681 with a p-value of 0.0000. The null hypothesis here would be that the model fits no better than a model with no predictors. The very low p-value (much less than 0.05) indicates that we reject the null hypothesis, confirming that the model provides a statistically significant improvement in fit.
Thus, the model is statistically significant in explaining the probability of car ownership as a function of log-income.
### a. Interpret the estimated logit model.
The estimated logit model is given by:
[tex]\[ \hat{L}_i = -2.77231 + 0.347582 \cdot \ln(\text{Income}) \][/tex]
Where:
- [tex]\(\hat{L}_i\)[/tex] is the estimated logit which is a linear combination of the coefficients.
- The coefficient for the intercept is -2.77231.
- The coefficient for the logarithm of income is 0.347582.
Interpretation:
- The intercept [tex]\(-2.77231\)[/tex] represents the log-odds of car ownership when the logarithm of income is zero [tex]\( \ln(\text{Income}) = 0 \)[/tex]. Since [tex]\(\ln(1) = 0\)[/tex], this essentially means the log-odds of car ownership when income is 1 unit of currency (assuming the unit used allows logarithm to make sense).
- The coefficient of [tex]\(\ln(\text{Income})\)[/tex], which is 0.347582, indicates how the log-odds of car ownership change with respect to a one-unit change in the logarithm of income.
- The t-values (in parentheses under the coefficients) indicate the statistical significance of these coefficients. The t-value for the intercept is -3.35, and for the slope (log of income) it is 4.05.
### b. From the estimated logit model, how would you obtain the expression for the probability of car ownership?
To convert the logit function into a probability, we use the logistic function. The probability [tex]\( \hat{P}_i \)[/tex] of car ownership can be obtained by transforming the logit [tex]\(\hat{L}_i\)[/tex]:
[tex]\[ \hat{P}_i = \frac{1}{1 + e^{-\hat{L}_i}} \][/tex]
Substituting [tex]\( \hat{L}_i \)[/tex] in, we get:
[tex]\[ \hat{P}_i = \frac{1}{1 + e^{-(-2.77231 + 0.347582 \cdot \ln(\text{Income}))}} \][/tex]
### c. What is the probability that a household with an income of 20,000 will own a car? And at an income level of 25,000? What is the rate of change of probability at the income level of 20,000?
Given the calculated values:
- For income of 20,000:
[tex]\[ \ln(\text{Income})_{20k} = 9.903487552536127 \][/tex]
[tex]\[ \text{logit}_{20k} = 0.6699640104856122 \][/tex]
[tex]\[ \text{probability}_{20k} = 0.6614951005017523 \][/tex]
- For income of 25,000:
[tex]\[ \ln(\text{Income})_{25k} = 10.126631103850338 \][/tex]
[tex]\[ \text{logit}_{25k} = 0.7475246923385082 \][/tex]
[tex]\[ \text{probability}_{25k} = 0.678639102980446 \][/tex]
- The rate of change of probability at the income level of 20,000 is:
[tex]\[ \text{rate of change}_{20k} = 0.07783032943385645 \][/tex]
### d. Comment on the statistical significance of the estimated logit model.
The model's statistical significance can be inferred from:
- The t-values for the intercept [tex]\(-2.77231\)[/tex] and the slope [tex]\(0.347582\)[/tex] are -3.35 and 4.05, respectively. A high t-value (in absolute terms) indicates that the coefficient is significantly different from zero. Given that the critical t-value for common significance levels (e.g., 1.96 for 5% significance) is much lower than the given t-values, both the intercept and slope coefficients are statistically significant.
- The chi-square ([tex]\( \chi^2 \)[/tex]) value for the goodness of fit is 16.681 with a p-value of 0.0000. The null hypothesis here would be that the model fits no better than a model with no predictors. The very low p-value (much less than 0.05) indicates that we reject the null hypothesis, confirming that the model provides a statistically significant improvement in fit.
Thus, the model is statistically significant in explaining the probability of car ownership as a function of log-income.
