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

    T --- A Substitute for Z

    Problem: We don't know Population Variability

    Solution: We assume that the sample's variability is a good basis for estimating the population's variability.

    • Recall, from Topic 5 , that the sample variance and sample standard deviation are unbiased estimates of the population variance and population standard deviation.

      The formula for the sample variance is:

      The formula for the sample standard deviation is:

    • Using these sample values, we can estimate the standard error of the distribution of sample means.
      As stated above (and developed in Topic 8 , the formula for the standard error is:

      The estimate of the standard error is simply:

    • Now, rather than calculating the Z-statistic using the (usually not) known population variance, we calcuate the T-Statistic by using the sample variance to estimate the standard error:

      Note the parallelism between Z and T
      Known Population Variance Unknown Population Variance

      The T-Statistic is used to test hypotheses about the population mean when the value for the population variance is unknown.
      The formula for the T-Statistic is similar in structure to that for the Z-Statistic, except that the T-Statistic uses estimated standard error rather than the (unknown) standard error.

    Next Topic: Using the T-Test
    Outline of this Lecture