Review of Hypothesis Testing, Confidence Intervals and Correlation
Type I and Type II Errors
-
Definition
-
A type I error has been committed if we do not accept the null hypothesis
(H0) when it is true. The probability of committing a type I error
is
, the level of significance of the test.
-
P(type I error) =
-
Definition
-
A type II error has been committed if we accept the null hypothesis
(H0) when it is false. The probability of committing a type II error
is denoted by
. This probability can be computed only
if a specific alternative hypothesis is available.
-
P(type II error) =
and are small.
Example
Suppose that we are testing the mean of a population. The level of
significance for the test is chosen to ,
and the population standard deviation is known to be 
.
Two specific hypotheses are given. The mean equals either
µ0 or µ1, where µ1 > µ0. We have
a sample of size n with
= 
,
where is normally distributed with mean
µ and standard deviation
.
The hypothesis test can be stated as:
Because, is a chosen value, and
µ1 > µ0, the critical value
z is obtained
from the standard normal distribution. The critical value of
= *,
that corresponds to z, is
calculated from
Now, the probability of type I error, with respect to the null hypothesis distribution, can be stated as:
Similarly, the probability of type II error, with respect to the alternative hypothesis distribution, can be stated as:
It can be noted, that for given values of ,
µ1, µ0,
and n, the critical value of
= *
is fixed. Now, if µ1 changes
closer to µ0 then increases,
because the alternative hypothesis distribution moves closer to the null hypothesis
distribution, and at the same time the shapes of the distributions remain unchanged.
If, on the other hand, µ1 changes and moves away from µ0
then decreases.
