The ROC curve shows how the true positive rate (sensitivity) changes as you vary the false positive rate (which is 1 minus specificity). A model that discriminates well will have a curve that sits toward the top-left of the plot, meaning you can achieve a high true positive rate with a low false positive rate. Key points to keep straight: the y-axis is sensitivity (true positive rate), not specificity; the x-axis is the false positive rate (1 − specificity), not sensitivity; and the 45-degree line represents random guessing, not perfect prediction. So the best reflection of strong performance is the curve hugging the upper-left corner, indicating high sensitivity with few false positives.

Forecast horizon is about how far into the future you’re predicting. As you extend that horizon, uncertainty grows. The further out you forecast, the more potential events, changes in trends, or shifts in drivers can occur, and you’re relying on patterns that may not hold. Errors can also compound because you often base longer-range forecasts on shorter-range projections, spreading the uncertainty downward through the forecast. That combination means predictions become less accurate the farther out you go. So the statement that forecast accuracy declines the further out you try to predict best describes the effect.

The ROC curve shows how well a binary classifier separates the two classes as you vary the decision threshold, by plotting true positive rate (sensitivity) against false positive rate (1 − specificity). The diagonal 45-degree line from (0,0) to (1,1) is the reference line for no discrimination. On this line, the true positive rate equals the false positive rate for every threshold, which is what happens with random guessing. In other words, the classifier has no real ability to distinguish the positive class from the negative class. The area under this line is 0.5, reflecting performance at chance level. So this diagonal line indicates no better than chance.

In a two-tailed z-test, you’re looking at how extreme the observed z-statistic is in either direction under the null hypothesis. The p-value is the probability of getting a value as extreme or more extreme in either tail. For a standard normal Z, that two-tailed p-value is the tail area beyond the absolute value of the observed z: p = P(|Z| ≥ |z|) = 2Φ(-|z|) = 2(1 − Φ(|z|)). If you only look at the probability that Z falls between −|z| and |z|, that central probability is Φ(|z|) − Φ(−|z|) = 2Φ(|z|) − 1, which is not the p-value in the standard definition; it represents the area inside the interval, not the combined tail area beyond it. So the conventional two-tailed p-value uses the tail regions and is 2Φ(−|z|). The expression 2Φ(z) − 1 corresponds to the central probability within ±z, not the p-value itself.