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

T-Statistic

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.