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Type I and Type II Errors

A test of significance leads us to one of two conclusions: either reject H0 or accept H0.
Terminology:
If we reject H0 when it is correct, we are said to have committed a Type I error .
If we accept H0 when it is false, we are said to have committed a Type II error. 
Recall the bowling example. Suppose that the bowler really does have an average that is 180 or higher, but `luck of the draw' led us to observe unfortunately low scores 125, 155 and 140. Based on the data, we wrongly reject his claim $H_0: \mu\geq 180$. This wrong decision is called a Type I error. Since our test procedure rejects H0 only when the observed test statistic is deep in H1 territory (in the extreme 5% territory, in fact), then the test procedure commits Type I error with less than .05 probability.

On the other hand, if the bowler really does have an average that is lower than 180, but the scores we observe were not low enough for us to reject his claim $H_0: \mu\geq 180$, then we have committed a Type II error.

Since the likelihood of Type I error is required to be below .05, the likelihood of committing Type II error may be large. The bowler's claim of $H_0: \mu\geq 180$ will be rejected only when the observed sample scores are extremely low (i.e. in the lowest 5% region). If we observed scores of 175, 178, 172, we cannot reject his claim, can we. Why not? Because a 180-bowler can very likely produce these scores! Thus we are forced to accept his claim, even though in truth he may only be a 175 bowler (in which case we have already committed Type II error).

Notes:
The likelihood of committing Type I error is less than .05.
The likelihood of committing Type II error may be quite large.
In scientific studies, the likelihood of Type II error is often reduced to desired levels by increasing the size of the observed sample. The required calculations are beyond the scope of this textbook.


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Next: Using the TI-83 Up: Test of Significance involving Previous: Examples

2003-09-08