| The process of hypothesis testing
when we don't know the population standard deviation
is the same as the process when we do know it, except
for two changes:
- The standard error is calculated differently.
- (Classical Approach) The critical region is
different. We need to know the degrees of freedom.
We have the same four steps:
Next Topic: T-Test Example
- State the Hypotheses:
This step is the same as with the Z-Statistic. We state
null and alternative hypotheses.
- (Classical Approach)
Set the decision criteria:
- Specify alpha, the significance level.
This is the same as with the Z-Statistic. For example:
- Determine the critical value of T.
Here there is a new complication in using T: There
isn't just one T-distribution that we use to determine
the critical value of T. There is a whole family of
distributions. The distribution depends on the "degrees-of-freedom",
which is simply equal to one less than the sample
size. That is:
We locate the critical T value by using the specified
alpha-level and df in the T distribution table in
The computer uses the T-distribution and the degrees-of-freedom
to calculate the exact probability of the result of the
- Gather Data.
This step is the same as with the Z-Statistic.
- Evaluate the Null Hypothesis.
We calculate the standard error of the mean using the
sample's standard deviation. Then we calculate T.
- Determine the standard error of the mean. The
standard error is calculated by the formula:
- Calculate the Test-Statistic. The T formula
- Make a decision:
- Classical Approach: We find the P value
for the obtained T and known df, comparing it to
the critical T value. If the obtained P is less
than alpha, we reject the null hypothesis.
- Contemporary Approach: The computer calculates
the P value. We report it and let the reader/listener
Outline of this Lecture