A verdict in a hypothesis test can be wrong in two different ways — and they are not the same mistake. Lay the truth against our decision:

• Type I error — we reject a true H_0. A false positive: we cry "effect!" when there is none. Its long-run rate is exactly the significance level, \alpha.
• Type II error — we fail to reject a false H_0. A false negative: a real effect slips past us. Its rate is written \beta.

The two correct verdicts are rejecting a false H_0 and failing to reject a true one.

Power: catching a real effect

If \beta is the chance of missing a real effect, then 1 - \beta is the chance of catching it. This is the power of the test:

A powerful test rarely lets a genuine effect through. So a good test wants both \alpha small (few false alarms) and power high (few misses).

The two errors, side by side

The left bell is the sampling distribution under H_0; the right bell is the distribution under H_1 (a real effect shifts the mean across). The vertical line is the decision threshold: we reject H_0 for anything to its right.

• The α region is the slice of the null bell to the right of the threshold — a true H_0 that we wrongly reject.
• The β region is the slice of the alternative bell to the left of the threshold — a false H_0 that we wrongly keep.

The trade-off

The two errors pull against each other. Slide the threshold right to shrink \alpha (fewer false positives) and the \beta slice of the alternative bell grows — more misses, less power. Slide it left and the trade reverses. With fixed sample size you cannot drive both errors to zero at once; choosing \alpha is choosing where on this see-saw to sit. (Collecting more data is what separates the bells and eases the squeeze.)