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Notes on Topic 9:
Z-Tests and T-Tests:
One Sample Hypothesis Tests

    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:
    1. The standard error is calculated differently.
    2. (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