College practice · Worked answers
Type I vs Type II error practice
The fastest way to sort Type I and Type II errors is to write the null hypothesis first. Then compare the decision with what is actually true. The words “false positive” and “false negative” help only after you decide what a positive result means in the problem.
Use a two-by-two decision table
| Decision | H₀ is true | H₀ is false |
|---|---|---|
| Reject H₀ | Type I error | Correct rejection; power |
| Fail to reject H₀ | Correct non-rejection | Type II error |
Type I means rejecting a true null. Type II means failing to reject a false null. Neither label can be assigned from the decision alone; the problem must tell you the underlying truth.
Put α, β, and power in the right place
α controls the chance of a Type I error under the null model. β is the chance of a Type II error at a specified alternative. Power = 1 − β, the chance of rejecting H₀ at that alternative. Power depends on more than α: sample size, variability, test design, and the effect being studied matter too.
With sample size, test, and effect held fixed, making the rejection rule stricter usually reduces Type I error but increases Type II error. A larger sample can improve power without raising α. There is no universal rule that one type of error is always worse.
Translate a story without changing the null
- Copy H₀ into a short sentence.
- Write what actually happened: is that sentence true or false?
- Write the test’s decision: reject or fail to reject?
- Find the matching table cell.
“Fail to reject” is deliberately cautious. It does not mean accept the null as a proven fact.
Work through two examples
A quality-control false alarm
- Let H₀ say a machine’s mean fill is on target. In reality, the mean is on target.
- The test rejects H₀ and flags the machine.
- It rejected a true null: Type I error. This is a false alarm under this definition of the null.
A missed change and its power
- Let H₀ say a teaching method has no effect. The method really does have the specific effect being studied, but the test fails to reject H₀.
- That is a Type II error: a false null was not rejected.
- If β for that specified effect is 0.18, power is 1 − 0.18 = 0.82, or 82%.
Try it yourself
Choose an answer, then check your reasoning. You can retry any question or open its solution.
Show solution 1
Type I error. Rejecting a true null is a Type I error. The underlying truth is what distinguishes it from a correct rejection.
Why the other choices are wrong:
- Type II error: Type II is failing to reject a false H₀. Here H₀ was true and got rejected.
- Correct decision: Rejecting a null that is actually true is an error.
- A correct rejection that shows the test’s power: Power is about rejecting a false H₀. This H₀ is true, so rejecting it is a mistake.
Show solution 2
Type II error. Failing to reject a false null is a Type II error. The test missed the specified departure from H₀.
Why the other choices are wrong:
- Type I error: Type I needs a rejection of a true H₀. Here nothing was rejected, and H₀ was false.
- Correct decision: H₀ was false, so failing to reject it missed a real effect.
- A false positive: A missed effect is usually called a false negative. A false positive would be rejecting a true H₀.
Show solution 3
0.75. Power = 1 − β = 1 − 0.25 = 0.75.
Why the other choices are wrong:
- 0.25: 0.25 is β, the Type II error rate. Power is 1 − β.
- 1.25: That adds instead of subtracting. A probability can’t be above 1.
- 0.95: 0.95 looks like 1 − α for α = 0.05. Power uses β, not α.
Show solution 4
A correct rejection. This is a correct rejection. Rejection is not automatically an error; it depends on whether H₀ is true.
Why the other choices are wrong:
- Type I error: Type I requires H₀ to be true. Here H₀ is false, so rejecting it is right.
- Type II error: Type II means failing to reject a false H₀. This test did reject it.
- You can’t tell without the p-value: The problem tells you the truth and the decision, which is all the table needs. The p-value doesn’t change the label.
Show solution 5
A 5% Type I error rate under H₀. α is a rejection-error rate under the null model. It is not the probability H₀ is true and does not determine power by itself.
Why the other choices are wrong:
- A 5% chance that H₀ is true: α is set assuming H₀ is true. It isn’t the probability that H₀ is true.
- A 95% chance of detecting every effect: That describes power, which depends on sample size, effect size, and more. Power isn’t 1 − α.
- A 5% chance of a Type II error: Type II error rate is β, which depends on the alternative. α is about Type I errors.
Show solution 6
Type I risk goes down and Type II risk goes up. A smaller α means fewer false rejections of a true null. With nothing else changed, it also makes a real effect harder to detect, so β rises and power falls.
Why the other choices are wrong:
- Both error risks go down: At a fixed sample size there is a trade-off. A stricter rule rejects less often, so more false nulls get missed.
- Type I risk goes up and Type II risk goes down: That is backward. A smaller α makes rejecting harder, which lowers Type I risk.
- Power goes up: Power goes down, because the test now needs stronger evidence to reject H₀.
Show solution 7
Use a larger sample. A larger sample shrinks the standard error, so a real effect of the same size is easier to detect while α stays the same.
Why the other choices are wrong:
- Lower α to 0.01: That makes rejecting harder, which lowers power.
- Use a smaller sample to cut down on noise: Fewer observations mean more sampling variability, so power goes down.
- Write “accept H₀” instead of “fail to reject H₀”: Changing the wording doesn’t change the procedure or its error rates.
Show solution 8
Type I error. The test rejected H₀ (“no disease”) even though H₀ was true. That is a Type I error, which is a false positive in this setup.
Why the other choices are wrong:
- Type II error: Type II would be missing a disease that is really there: failing to reject a false H₀.
- Correct decision: The patient is healthy, so flagging them is a mistake.
- Type II, because the wrong result could harm the patient: The label comes from comparing the truth with the decision, not from how costly the mistake is.
Answer key
Open all answers
- Type I error. Rejecting a true null is a Type I error. The underlying truth is what distinguishes it from a correct rejection.
- Type II error. Failing to reject a false null is a Type II error. The test missed the specified departure from H₀.
- 0.75. Power = 1 − β = 1 − 0.25 = 0.75.
- A correct rejection. This is a correct rejection. Rejection is not automatically an error; it depends on whether H₀ is true.
- A 5% Type I error rate under H₀. α is a rejection-error rate under the null model. It is not the probability H₀ is true and does not determine power by itself.
- Type I risk goes down and Type II risk goes up. A smaller α means fewer false rejections of a true null. With nothing else changed, it also makes a real effect harder to detect, so β rises and power falls.
- Use a larger sample. A larger sample shrinks the standard error, so a real effect of the same size is easier to detect while α stays the same.
- Type I error. The test rejected H₀ (“no disease”) even though H₀ was true. That is a Type I error, which is a false positive in this setup.
Common questions
Is Type I always worse than Type II?
No. The consequences depend on the setting. State the null and describe both errors before judging their costs.
Can I tell which error happened from a p-value?
Usually not. You know the decision, but the true population state is generally unknown.
What does 80% power mean?
At a specified alternative and under the design assumptions, the procedure rejects the null in about 80% of repeated studies.
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Sources and fact checks
Definitions and methods checked against these textbook references. Practice scenarios and worked numbers were written for this guide.