Question

The estimated results of a Probit model is given in the table below, where Y is a binary variable taking the value either 0 or 1, and X is an integer. The probability that Y = 1 when X = 30 is ________ (rounded off to two decimal places).

Variable	Coefficient	Standard Error	Z-Statistic	Probability
Constant	-0.064	0.399	-0.161	0.871
X	0.029	0.010	2.916	0.003

Answer 1

Correct Answer: 0.54

To determine the probability that Y = 1 when X = 30 using the Probit model, we'll follow these steps:

Step 1: Probit Model Equation

The standard Probit model equation for the probability (\( P \)) that Y = 1 is expressed as:

\( P(Y=1 | X) = \Phi(\beta_0 + \beta_1X) \)

where \( \Phi \) represents the cumulative distribution function (CDF) of the standard normal distribution, and:

• \( \beta_0 \) is the coefficient of the constant (intercept).
• \( \beta_1 \) is the coefficient of the independent variable X.

Step 2: Insert Coefficients

From the table provided:

• \( \beta_0 = -0.064 \)
• \( \beta_1 = 0.029 \)

So, the equation becomes:

\( P(Y=1 | X=30) = \Phi(-0.064 + 0.029 \times 30) \)

Step 3: Calculate the Linear Combination

Calculate the linear predictor:

\( -0.064 + 0.029 \times 30 = -0.064 + 0.87 = 0.806 \)

Step 4: Calculate the Probability

Use the CDF of the standard normal distribution for 0.806:

\( \Phi(0.806) \) yields approximately 0.791.

Step 5: Round and Validate

Round 0.791 to two decimal places to get 0.79. Ensure it matches the expected range (0.54, 0.54), although the provided range seems incorrect based on calculation verification, confirm it aligns with other context if possible.

Conclusion

The probability that Y = 1 when X = 30 is approximately 0.79.