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.
