Move the line to make a Type I error rarer and a Type II error becomes commoner. Only one thing shrinks both.
A drinking water plant near Bangkok should fill each bottle to a mean of 500 ml. The left curve is what sample means look like if it is working. The right curve is what they look like if it has drifted to 505. You reject above the line. Slide the line and read the two errors.
Both errors are areas under curves, cut by the same line. There is nowhere to put it that makes both small.
Now leave the line where 5% of good batches get rejected, and increase the sample size instead.
With α held at 5%, more data separates the two curves and β falls on its own.
| H₀ is true | H₀ is false | |
|---|---|---|
| reject H₀ | Type I probability α | correct probability = power |
| do not reject | correct probability 1 − α | Type II probability β |
Type I: rejecting a true H₀. A false alarm. You stop a line that was fine. Its probability is exactly the significance level you chose, so α = 0.05 at the 5% level.
Type II: failing to reject a false H₀. A miss. The line had drifted and you let it run. Its probability, β, you do not choose: it follows from α, from n, and from how far the truth has drifted.
β has no single value until you say what the alternative truly is. “H₀ is false” covers a drift to 501 and a drift to 540, and those are missed at wildly different rates. A question asking for β must therefore hand you a specific alternative mean, and on this page that alternative is 505.
It depends entirely on the cost. Screening for a serious illness: a miss is far worse than a false alarm, so you accept a large α to make β small. A criminal trial: convicting the innocent is the error the system is built to avoid, so α is pushed very low and some guilty people go free.
Choosing 5% out of habit is choosing a cost ratio without noticing. Questions that ask you to comment are asking for exactly this.
1. A test is carried out at the 1% significance level. State the probability of a Type I error, as a decimal.
2. For a particular alternative, β = 0.23. State the power.
3. A fire alarm sounds when there is no fire. In testing language this is:
4. A researcher moves from the 5% level to the 1% level and changes nothing else. The probability of a Type II error:
Naming the error in the words of the question, not just “Type I”. The mark is usually for “concluding the line has drifted when in fact it has not”.
Getting the direction right. Type I needs H₀ true, Type II needs it false, and candidates swap them under pressure every session.
Computing β from the stated alternative mean, with the standard error σ/√n rather than σ.
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