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Understand Type I and Type II errors, their probabilities, and the concept of power.
Learn step-by-step with practice exercises built right in.
A Type I error occurs when we reject a true null hypothesis.
In words: We conclude there's an effect/difference when actually there isn't one.
Probability of Type I error = α (the significance level)
Example:
Define Type I and Type II errors. In the context of a legal trial, explain what each would mean.
Type I error: Rejecting when is true. Probability is . Type II error: Failing to reject when is true. Probability is . In a trial: = defendant is innocent, = defendant is guilty. Type I: Convicting an innocent person (rejecting innocent/true ). Type II: Acquitting a guilty person (failing to reject innocent when guilt is true). Society typically considers Type I more serious, setting strict conviction standards (low ).
Avoid these 3 frequent errors
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In medical testing:
A Type II error occurs when we fail to reject a false null hypothesis.
In words: We conclude there's no effect/difference when actually there is one.
Probability of Type II error = β (often unknown)
Example:
In medical testing:
The power of a test is the probability of correctly rejecting when is true.
Interpretation: If the alternative is true, power is the probability we'll find it.
| True | False | |
|---|---|---|
| Reject | Type I Error (prob = α) | Correct (prob = power) |
| Fail to Reject | Correct (prob = 1 − α) | Type II Error (prob = β) |
Scenario: Testing if a new drug is effective.
Type I error (α = 0.05): Conclude drug works when it doesn't. Risk: 5% (set by significance level)
Type II error (β): Conclude drug doesn't work when it actually does. Risk: Unknown, but reduced by:
Power = 1 − β: Probability we detect the drug's effect if it exists. Should be high (0.80 or 0.90 typical targets)
| Factor | Effect on α | Effect on β |
|---|---|---|
| Increase α (e.g., 0.05 → 0.10) | Increases Type I risk | Decreases Type II risk |
| Increase n (sample size) | No effect | Decreases (higher power) |
| Increase effect size | No effect | Decreases (easier to detect) |
| Increase confidence (reduce α) | Decreases | Increases (lower power) |
Lowering α (e.g., from 0.05 to 0.01) automatically increases β. You can't simultaneously minimize both errors with fixed sample size.
Strategy depends on consequence:
To increase power for fixed α:
Example: If power is too low (say 0.60), increasing n to 200 might increase power to 0.85.
Know the definitions of Type I and II errors cold; these appear frequently. Context matters: identify which error is worse, then design accordingly. Power problems ask: "What sample size gives power = 0.90?" Use technology or power tables. Always interpret in context (what does rejecting/failing to reject mean for the actual situation?).
A medical test for a disease has and . Interpret each in context and compute the power.
: If a person does NOT have the disease (true negative), there is a 1% chance the test incorrectly says they do (false positive). : If a person DOES have the disease (true positive condition), there is a 10% chance the test fails to detect it (false negative). Power = . This means the test correctly identifies disease presence 90% of the time when disease is present. A higher power is desirable for medical tests.
Describe two practical factors that affect the power of a hypothesis test and explain how each influences power.
Factor 1 — Sample size (): Larger increases power because it reduces standard error, making it easier to detect true differences. A large sample produces a narrower sampling distribution, so the test statistic is more extreme when is true. Factor 2 — Significance level (): Larger increases power. Setting instead of makes it easier to reject , but increases the Type I error rate. There is a trade-off between power and Type I error control. Additional factor: Effect size (the true difference from hypothesized value). Larger true effect → higher power.