In AP Statistics, the power of a test is the probability that a significance test correctly rejects a false null hypothesis. Power equals 1 − β, where β is the probability of a Type II error, so higher power means the test is better at detecting a real effect when one exists.
Power is your test's ability to catch a real effect. Formally, it's the probability that a significance test rejects the null hypothesis given that the null hypothesis is actually false. Think of it like a smoke detector. A high-power detector goes off when there's a real fire. A low-power detector lets the fire burn while telling you everything is fine.
Power is mathematically locked to Type II error. A Type II error happens when the null is false but you fail to reject it (a false negative), and its probability is called β. The CED states the relationship directly. Power = 1 − β, or equivalently, P(Type II error) = 1 − power. So if a study has power 0.72 to detect a real improvement, there's a 0.28 chance it misses that improvement entirely. The good news for the exam is that you don't have to compute power from scratch. You need to define it, convert between power and β, and explain what makes power go up or down.
Power lives in Topic 6.7 (Potential Errors When Performing Tests) in Unit 6, under learning objectives 6.7.B and 6.7.C. LO 6.7.B says you should know that power is the probability a test correctly rejects a false null and that P(Type II error) = 1 − power. LO 6.7.C asks you to identify what changes power. The CED lists four levers that decrease Type II error (and therefore increase power): a larger sample size, a higher significance level α, a smaller standard error, and a true parameter value farther from the null.
Here's the big-picture payoff. Power is where statistics meets real decisions. A clinical trial with low power can run a perfectly valid test and still miss a drug that actually works. That's why exam questions love asking 'how could the researchers increase the power of this test?' The expected answers come straight from that CED list.
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Type II Error (Unit 6)
Power and Type II error are two sides of the same coin. β is the probability of missing a real effect, and power = 1 − β is the probability of catching it. If you know one, you instantly know the other.
Type I Error (Unit 6)
There's a built-in trade-off. Raising α (the Type I error rate) makes it easier to reject the null, which increases power but also increases your chance of a false positive. You can't crank both error rates down for free without other changes.
Sample Size (Units 1 & 5-9)
Sample size is the one lever that improves power with no downside to error rates. A bigger n shrinks the standard error, which makes real effects easier to detect. This is why 'increase the sample size' is almost always a correct answer to 'how do we increase power?'
Alternative Hypothesis (Units 6-9)
Power is always calculated assuming the alternative hypothesis is true. It also depends on which specific alternative value you're trying to detect. A true mean far from the null value is easy to spot (high power), while one barely different from the null is hard to spot (low power).
Power shows up mostly in two flavors. First, the quick conversion. Multiple-choice questions give you β and ask for power, or give you power and ask for the probability of failing to detect a real effect. If power is 0.72, the chance of missing the true effect is 0.28. Done. Second, the conceptual question. You'll see a scenario (often a drug trial or quality-control setting) and get asked which change increases power, or you'll be asked to suggest one on an FRQ. Acceptable answers come straight from the CED: increase the sample size, increase α, decrease the standard error, or have a true parameter value farther from the null (though researchers can't control that last one). One trap to avoid is mixing up the conditioning. Power is a probability computed assuming the null is false. If a question describes failing to reject a false null, that's a Type II error, and its probability is 1 − power.
These are complements, not the same thing. β is the probability of a Type II error, meaning the test fails to reject a null hypothesis that's actually false. Power is the probability the test succeeds in rejecting that false null. Power = 1 − β always. So when a question says 'the probability of failing to detect a true effect,' it's asking about β, and when it says 'the probability of correctly detecting a true effect,' it's asking about power. Both quantities only make sense when the alternative hypothesis is true.
Power is the probability that a significance test correctly rejects a false null hypothesis.
Power = 1 − β, so the probability of a Type II error is always 1 minus the power of the test.
Power increases when sample size increases, when α increases, when standard error decreases, or when the true parameter value is farther from the null.
Increasing α raises power but also raises the chance of a Type I error, so there's a trade-off; increasing sample size raises power without that cost.
Power is calculated under the assumption that the alternative hypothesis is true, never under the null.
On the AP exam you won't compute power from scratch; you'll convert between power and β and explain how to increase power.
Power is the probability that a hypothesis test correctly rejects a false null hypothesis. In other words, it's the chance your test detects a real effect when one actually exists. It's defined in Topic 6.7 under learning objective 6.7.B.
No. 1 − α is the probability of correctly failing to reject a true null hypothesis. Power is 1 − β, the probability of correctly rejecting a false null. α deals with Type I error, while power deals with Type II error, and they're calculated under opposite assumptions about the null.
They're complements. A Type II error (probability β) means failing to reject a false null, while power (1 − β) means correctly rejecting it. If a test has power 0.72, the probability of a Type II error is 0.28.
Not from scratch. The AP exam expects you to convert between power and β using power = 1 − β, interpret power in context, and identify the four factors that increase it (larger sample size, larger α, smaller standard error, true value farther from the null).
Increase the sample size. A bigger sample shrinks the standard error, making real effects easier to detect, and unlike raising α, it doesn't increase your Type I error rate. The true parameter's distance from the null also affects power, but researchers can't control that.