Type I and Type II Errors - Tests of Hypotheses | Biostatistics

**TYPE I AND TYPE II ERRORS**

In Section 9.1, we defined the type I error α as the probability of rejecting the null hypothesis when the null
hypothesis is true. We saw that in the Neyman–Pearson formulation of hypothesis
testing, the type I error rate is fixed at a certain low level. In practice,
the choice is usually 0.05 or 0.01. In Sections 9.3 through 9.5, we saw
examples of how critical regions were defined based on the distribution of the
test statistic under the null hypothesis.

Also in Section 9.1, we defined the type II error
as β. The type II error is the probability of not rejecting the null
hypothesis when the null hypothesis is false. It depends on the “true” value of
the parameter under the alternative hypothesis.

For example, suppose we are testing a null hypothesis
that the population mean μ = μ_{0}. The type II error depends on the value of μ = μ_{1} ≠ μ_{0} under the alternative hypothesis. In the next section, we see that the
power of a test is defined as 1 – β. The
term “power” refers to the probability of correctly rejecting the null
hypothesis when it is in fact false. Given that β depends
on the value of μ_{1} in the context of testing for a population mean,
the power is a function of μ_{1}; hence, we refer to a power function rather than a single number.

In sample size determination (Section 9.13), we will
see that analogous to choosing a width *d*
for a confidence interval, we will select a distance δ for | μ_{1} – μ_{0}| such that we achieve a specific high value for the power at that δ. Usually, the value for 1 -β is
chosen to be 0.80, 0.90, or 0.95.

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