Hypothesis Testing: T-Test

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:

1. State the Hypotheses:
   This step is the same as with the Z-Statistic. We state null and alternative hypotheses.

2. (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 Appendix.

   (Contemporary Approach)
   The computer uses the T-distribution and the degrees-of-freedom to calculate the exact probability of the result of the experiment.

3. Gather Data.
   This step is the same as with the Z-Statistic.

4. 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 is:
     
   • 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 decide.

Next Topic: T-Test Example
Outline of this Lecture