A reliable method
State the parameter and hypotheses, check conditions, standardize the observed difference, and compare the p-value with the chosen significance level. Report a contextual conclusion.
The common trap
A small p-value is not the probability that the null hypothesis is true. Failure to reject means insufficient evidence for the alternative; it does not establish equality.
Make the connection
A Type I error rejects a true null hypothesis. A Type II error fails to reject a false null. Statistical significance and a practically important effect are different questions.
Follow a hypothesis all the way to a claim
With null proportion .50, observed proportion .60 and null SE .05, z=2. This measures distance under the null; it is not itself a p-value. The alternative determines which tail or tails matter. After calculating a p-value, compare it with a threshold chosen independently of the observed result.
An error is a truth-decision combination
Rejecting a true null is Type I. Failing to reject a false null is Type II. When p=.04 and alpha=.03, fail to reject under the stated rule. That conclusion does not prove the null or imply that a practical effect is impossible. Statistical and practical importance are separate judgments.
Transfer check
Rewrite “p=.02 means a 2% chance the null is true.” A defensible version conditions on the null and describes results at least as extreme as those observed. Then explain why increasing sample size can detect smaller effects even when the practical importance of those effects is unchanged.
For a one-proportion test, p0 = 1/2, observed proportion = 11/20, and the null standard error is 1/20. What is z?
- Subtract the null proportion from the sample proportion.
- Divide by the null standard error.
- z = 1.
Compare the other answer choices
Divide by estimated variance instead of standard error.
Omit division by standard error.
Multiply by standard error instead of dividing.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Statistics 2026–27 course framework