A financial institution is using a logistic regression model to predict the likelihood of a loan default based on various borrower characteristics. To make predictions using the model, a threshold value Z is chosen, and the predicted probability $p_i$ is compared to the threshold. Which of the following statements is most likely accurate? If $p_i$ is:

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Explanation:

Explanation

In logistic regression models for binary classification (such as predicting loan default), a threshold value Z is used to convert predicted probabilities into discrete outcomes.

Correct Logic:

When the predicted probability $p_i$ is greater than or equal to the threshold Z, the model predicts the positive outcome (default = 1)
When the predicted probability $p_i$ is less than the threshold Z, the model predicts the negative outcome (no default = 0)

This can be mathematically expressed as:

1 & \text{if } p_i \geq Z \\ 0 & \text{if } p_i < Z \end{cases}$$ **Why other options are incorrect:** - **Option B**: If $p_i$ is less than or equal to Z, the model would predict default, which is incorrect - this would lead to too many false positives - **Option C**: Using only "greater than" (without "equal to") would miss cases where $p_i$ exactly equals the threshold - **Option D**: If $p_i$ is less than Z, the model should predict no default, not default The threshold Z represents the cutoff probability above which the model classifies an observation as belonging to the positive class (default in this case).

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