Answer :
### Solution:
To address the given questions, we need to systematically interpret the logit model, derive necessary expressions, and use given statistical results.
### a. Interpret the estimated logit model.
The logit model tried to determine car ownership ([tex]\(Y\)[/tex]) as a function of income expressed logarithmically. The estimated logit model given is:
[tex]\[ \hat{L}_i = -2.77231 + 0.347582 \ln(\text{Income}) \][/tex]
where [tex]\(\hat{L}_i\)[/tex] is the logit value, which is a linear function of the natural logarithm of income.
- The intercept is [tex]\(-2.77231\)[/tex], which represents the log-odds of car ownership when the logarithm of income is zero.
- The coefficient [tex]\(0.347582\)[/tex] associated with [tex]\(\ln(\text{Income})\)[/tex] indicates that, holding all else constant, a one-unit increase in the natural logarithm of income will increase the log-odds of car ownership by approximately [tex]\(0.347582\)[/tex].
The t-values provided ([tex]\(-3.35\)[/tex] for the intercept and [tex]\(4.05\)[/tex] for [tex]\(\ln(\text{Income})\)[/tex]) suggest the statistical significance of the coefficients since they are substantially larger than 2 in absolute value.
### b. From the estimated logit model, how would you obtain the expression for the probability of car ownership?
The probability of car ownership ([tex]\(P(Y = 1)\)[/tex]) can be obtained using the logistic function:
[tex]\[ P(Y = 1) = \frac{1}{1 + e^{-\hat{L}_i}} \][/tex]
Substituting [tex]\(\hat{L}_i\)[/tex] into this equation:
[tex]\[ \hat{P}(Y = 1) = \frac{1}{1 + e^{-( -2.77231 + 0.347582 \ln(\text{Income}))}} \][/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?
To find the probabilities, we substitute the given incomes into the probability equation.
For an income of 20,000:
1. Calculate the logit:
[tex]\[ \hat{L}_{20,000} = -2.77231 + 0.347582 \ln(20000) = 0.6699640104856122 \][/tex]
2. Convert the logit to probability using the logistic function:
[tex]\[ P_{20,000} = \frac{1}{1 + e^{-(0.6699640104856122)}} = 0.6614951005017523 \][/tex]
For an income of 25,000:
1. Calculate the logit:
[tex]\[ \hat{L}_{25,000} = -2.77231 + 0.347582 \ln(25000) = 0.7475246923385082 \][/tex]
2. Convert the logit to probability using the logistic function:
[tex]\[ P_{25,000} = \frac{1}{1 + e^{-(0.7475246923385082)}} = 0.678639102980446 \][/tex]
Rate of change of probability at the income level of 20,000:
To find the rate of change of the probability with respect to income, we need the derivative of the probability:
1. Partial derivative of the logit with respect to income:
[tex]\( \frac{\partial \hat{L}}{\partial \text{Income}} = \frac{0.347582}{\text{Income}} \)[/tex]
At income of 20,000:
[tex]\( \frac{\partial \hat{L}}{\partial \text{Income}} = \frac{0.347582}{20000} \)[/tex]
2. Partial derivative of the probability with respect to the logit:
[tex]\( \frac{\partial \hat{P}}{\partial \hat{L}} = \hat{P}(1 - \hat{P}) \)[/tex]
At [tex]\(\hat{P}_{20,000} = 0.6614951005017523\)[/tex]:
[tex]\( \frac{\partial \hat{P}}{\partial \hat{L}} = 0.6614951005017523 (1 - 0.6614951005017523) \)[/tex]
3. Rate of change:
Multiplying these partial derivatives:
[tex]\( \frac{d\hat{P}}{d \text{Income}} = \hat{P}(1 - \hat{P}) \times \frac{0.347582}{20000} = 3.891516471692822 \times 10^{-6} \)[/tex]
### d. Comment on the statistical significance of the estimated logit model.
The statistical significance of the estimated logit model is evaluated by looking at the t-values and the chi-square statistic provided:
- The t-values for the intercept ([tex]\(-3.35\)[/tex]) and the coefficient of [tex]\(\ln(\text{Income})\)[/tex] ([tex]\(4.05\)[/tex]) are large in magnitude (greater than 2), indicating that both the intercept and the coefficient are significantly different from zero at conventional significance levels (typically 5% or 1%). Hence, [tex]\( \ln(\text{Income}) \)[/tex] significantly impacts the log-odds of car ownership.
- The chi-square statistic ([tex]\( x^2(1 \text{ df })=16.681 \)[/tex]) with a p-value of [tex]\(0.0000\)[/tex] signifies a very strong rejection of the null hypothesis that the explanatory variable ([tex]\(\ln(\text{Income})\)[/tex]) does not contribute to the model. This indicates that the model fits the data significantly better than a model with no predictors.
In summary, the logit model is statistically significant and suggests that income, when expressed logarithmically, plays a crucial role in predicting car ownership among households.
To address the given questions, we need to systematically interpret the logit model, derive necessary expressions, and use given statistical results.
### a. Interpret the estimated logit model.
The logit model tried to determine car ownership ([tex]\(Y\)[/tex]) as a function of income expressed logarithmically. The estimated logit model given is:
[tex]\[ \hat{L}_i = -2.77231 + 0.347582 \ln(\text{Income}) \][/tex]
where [tex]\(\hat{L}_i\)[/tex] is the logit value, which is a linear function of the natural logarithm of income.
- The intercept is [tex]\(-2.77231\)[/tex], which represents the log-odds of car ownership when the logarithm of income is zero.
- The coefficient [tex]\(0.347582\)[/tex] associated with [tex]\(\ln(\text{Income})\)[/tex] indicates that, holding all else constant, a one-unit increase in the natural logarithm of income will increase the log-odds of car ownership by approximately [tex]\(0.347582\)[/tex].
The t-values provided ([tex]\(-3.35\)[/tex] for the intercept and [tex]\(4.05\)[/tex] for [tex]\(\ln(\text{Income})\)[/tex]) suggest the statistical significance of the coefficients since they are substantially larger than 2 in absolute value.
### b. From the estimated logit model, how would you obtain the expression for the probability of car ownership?
The probability of car ownership ([tex]\(P(Y = 1)\)[/tex]) can be obtained using the logistic function:
[tex]\[ P(Y = 1) = \frac{1}{1 + e^{-\hat{L}_i}} \][/tex]
Substituting [tex]\(\hat{L}_i\)[/tex] into this equation:
[tex]\[ \hat{P}(Y = 1) = \frac{1}{1 + e^{-( -2.77231 + 0.347582 \ln(\text{Income}))}} \][/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?
To find the probabilities, we substitute the given incomes into the probability equation.
For an income of 20,000:
1. Calculate the logit:
[tex]\[ \hat{L}_{20,000} = -2.77231 + 0.347582 \ln(20000) = 0.6699640104856122 \][/tex]
2. Convert the logit to probability using the logistic function:
[tex]\[ P_{20,000} = \frac{1}{1 + e^{-(0.6699640104856122)}} = 0.6614951005017523 \][/tex]
For an income of 25,000:
1. Calculate the logit:
[tex]\[ \hat{L}_{25,000} = -2.77231 + 0.347582 \ln(25000) = 0.7475246923385082 \][/tex]
2. Convert the logit to probability using the logistic function:
[tex]\[ P_{25,000} = \frac{1}{1 + e^{-(0.7475246923385082)}} = 0.678639102980446 \][/tex]
Rate of change of probability at the income level of 20,000:
To find the rate of change of the probability with respect to income, we need the derivative of the probability:
1. Partial derivative of the logit with respect to income:
[tex]\( \frac{\partial \hat{L}}{\partial \text{Income}} = \frac{0.347582}{\text{Income}} \)[/tex]
At income of 20,000:
[tex]\( \frac{\partial \hat{L}}{\partial \text{Income}} = \frac{0.347582}{20000} \)[/tex]
2. Partial derivative of the probability with respect to the logit:
[tex]\( \frac{\partial \hat{P}}{\partial \hat{L}} = \hat{P}(1 - \hat{P}) \)[/tex]
At [tex]\(\hat{P}_{20,000} = 0.6614951005017523\)[/tex]:
[tex]\( \frac{\partial \hat{P}}{\partial \hat{L}} = 0.6614951005017523 (1 - 0.6614951005017523) \)[/tex]
3. Rate of change:
Multiplying these partial derivatives:
[tex]\( \frac{d\hat{P}}{d \text{Income}} = \hat{P}(1 - \hat{P}) \times \frac{0.347582}{20000} = 3.891516471692822 \times 10^{-6} \)[/tex]
### d. Comment on the statistical significance of the estimated logit model.
The statistical significance of the estimated logit model is evaluated by looking at the t-values and the chi-square statistic provided:
- The t-values for the intercept ([tex]\(-3.35\)[/tex]) and the coefficient of [tex]\(\ln(\text{Income})\)[/tex] ([tex]\(4.05\)[/tex]) are large in magnitude (greater than 2), indicating that both the intercept and the coefficient are significantly different from zero at conventional significance levels (typically 5% or 1%). Hence, [tex]\( \ln(\text{Income}) \)[/tex] significantly impacts the log-odds of car ownership.
- The chi-square statistic ([tex]\( x^2(1 \text{ df })=16.681 \)[/tex]) with a p-value of [tex]\(0.0000\)[/tex] signifies a very strong rejection of the null hypothesis that the explanatory variable ([tex]\(\ln(\text{Income})\)[/tex]) does not contribute to the model. This indicates that the model fits the data significantly better than a model with no predictors.
In summary, the logit model is statistically significant and suggests that income, when expressed logarithmically, plays a crucial role in predicting car ownership among households.
