Question at position 6 In a logistic regression model of customer churn (unsubscribing), a positive coefficient for a price means that as that price increases, the predicted probability of customer churn always increases.TrueFalse单项选择题
A
True
B
False

Question

Question at position 6 In a logistic regression model of customer churn (unsubscribing), a positive coefficient for a price means that as that price increases, the predicted probability of customer churn always increases.TrueFalse单项选择题

A

True

B

False

BlackTom AI · Accepted Answer

The logistic model uses the logistic (sigmoid) function p = 1 / (1 + e^(-η)), where η = β0 + β_price·price + (other terms). Because the sigmoid is a strictly increasing function of η, the sign of β_price determines whether p rises or falls as price increases (holding other covariates constant). 1) Relevant knowledge points and concepts - Logistic regression models the log-odds (η) as a linear function of predictors: η = β0 + β1·x + .... - The predicted probability p = σ(η) = 1/(1+e^{-η}) is a monotonic increasing function of η. - The derivative dp/dx = (dp/dη)·(dη/dx) = β1 · p(1 − p). Since 0 < p < 1, p(1 − p) > 0 for finite η. 2) Complete reasoning step by step - If β_price > 0 then dη/d(price) = β_price > 0. - dp/d(price) = β_price · p(1 − p). Because p(1 − p) > 0, dp/d(price) > 0 whenever β_price > 0. - A positive derivative means p strictly increases as price increases (for any finite change in price, holding other variables fixed). 3) Calculations with intermediate steps (symbolic and numeric example) - Symbolic: dp/dprice = β_price · p(1 − p). For β_price = 0.5 and p =, say, 0.2, dp/dprice = 0.5 · 0.2 · 0.8 = 0.08 > 0. - Numeric example: β0 = −2, β_price = 0.5. - At price = 0: η = −2 ⇒ p = 1/(1+e^{2}) ≈ 0.1192. - At price = 1: η = −1.5 ⇒ p = 1/(1+e^{1.5}) ≈ 0.1824. p increased. 4) Why the correct answer is correct - The logistic link is strictly increasing in η; a positive coefficient increases η as price increases, which increases the predicted probability. There is no non-monotonic behavior introduced by the logistic transform. 5) Why the other option (False) is incorrect - Saying the predicted probability does not always increase contradicts the monotonicity shown above. Only if β_price ≤ 0 would increases in price not increase the predicted probability. Interaction terms or simultaneous changes in other covariates could alter the net effect, but the statement implicitly considers the marginal effect holding others fixed. 6) Conclusion - Therefore the correct answer is True, which matches the provided answer.

Question at position 6 In a logistic regression model of customer churn (unsubscribing), a positive coefficient for a price means that as that price increases, the predicted probability of customer churn always increases.TrueFalse单项选择题

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