Given our existing work on confidence intervals, hypothesis testing will be relatively easy to develop.
Hypothesis testing is concerned with making decisions using data.
A null hypothesis is specified that represents the status quo, usually labeled \(H_0\)
The null hypothesis is assumed true and statistical evidence is required to reject it in favor of a research or alternative hypothesis.
A respiratory disturbance index of more than \(30\) events / hour, say, is considered evidence of severe sleep disordered breathing (SDB).
Suppose that in a sample of \(100\) overweight subjects with the risk factors for sleep disordered breathing at a sleep clinic, the mean RDI was \(32\) events / hour with a standard deviation of \(10\) events / hour
We might to test the hypothesis that
\(H_0\) : \(\mu = 30\)
\(H_a\) : \(\mu > 30\)
where \(\mu\) is the population mean RDI.
The alternative hypotheses are typically of the form \(\lt\), \(\gt\) or \(\ne\)
Note that there are four possible outcomes of our statistical decision process
| TRUTH | DECIDE | RESULT |
|---|---|---|
| \(H_0\) | \(H_0\) | Correctly accept null |
| \(H_0\) | \(H_a\) | Type I error |
| \(H_a\) | \(H_a\) | Correctly reject null |
| \(H_a\) | \(H_0\) | Type II error |
The Type I error and the Type II error are related in the sense that as the Type I error rate increases, the Type II error decreases, and vice versa.
Here, rejecting the null hypothesis is to convict the defendant.
If we set a very low standard, i.e., we don’t require much in evidence to convict people, then we’re going to increase the percentage of innocent people convicted, Type I errors in this case. However, we would also increase the percentage of guilty people convicted, which would be correctly rejecting the null hypothesis.
If we set a very high standard, basically a person has to have a smoking gun in their hand, to convict them, then we would increase the percentage of innocent people, a good thing, correctly accepting the null. But we would also increase the percentage of guilty people let free, or so called, Type II errors.
Of course ideally, what you would like is to get better evidence for a given standard. That’s the idea of doing things like increasing the sample size.