Committing an Error Section
Every time we make a decision and come to a conclusion, we must keep in mind that our decision is based on probability. Therefore, it is possible that we made a mistake.
Consider the example of the previous Lesson on whether the majority of Penn State students like the cold. In that example, we took a random sample of 500 Penn State students and found that 278 like the cold. We rejected the null hypothesis, at a significance level of 5% with a p-value of 0.006.
- Type I Error
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Rejecting \(H_0\) when \(H_0\) is really true, denoted by \(\alpha\) ("alpha") and commonly set at .05
\(\alpha=P(Type\;I\;error)\)
The significance level of 5% means that we have a 5% chance of committing a Type I error. That is, we have 5% chance that we rejected a true null hypothesis.
- Type II Error
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Failing to reject \(H_0\) when \(H_0\) is really false, denoted by \(\beta\) ("beta")
\(\beta=P(Type\;II\;error)\)
If we failed to reject a null hypothesis, then we could have committed a Type II error. This means that we could have failed to reject a false null hypothesis.
| Decision | Reality | ||
|---|---|---|---|
| \(H_0\) is true | \(H_0\) is false | Probability Level | |
| Reject \(H_0\), (conclude \(H_a\)) | Type I error | Correct decision | P is LESS than .05 that the null is true Small probabilities (less than .05) lead to rejecting the null |
| Fail to reject \(H_0\) | Correct decision | Type II error | P is GREATER than .05 that the null is true Large probability (greater than .05) lead to not rejecting the null |
How Important are the Conditions of a Test? Section
In our six steps in hypothesis testing, one of them is to verify the conditions. If the conditions are not satisfied, we cannot, however, make a decision or state a conclusion. The conclusion is based on probability theory.
If the conditions are not satisfied, there are other methods to help us make a conclusion. The conclusion, however, may be based on other parameters, such as the median. There are other tests (called nonparametric) that can be used.