The One-Sample t Test
Just like z scores, t scores can be used to conduct tests of significance.
H
0
is the same when conducting tests of significance using t scores as it is
when using z scores. It states that the sample mean is equal to the
population mean, or that there is no significant difference between the two
values.
The alternative hypothesis is also formulated in the same way. It can be one-
sided, stating that the sample mean is either higher or lower than the
population mean, or it can be two-sided, stating that the sample mean is
different (not necessarily higher or lower) that the population, which means
that the test is looking for very high or very low values.
Similarly, the formula for the t test statistic stays the same as for the z
statistic. The only difference is that the sample standard deviation, rather
than the population standard deviation, is used. Also, the value of the t
statistic differs based on sample size, or degrees of freedom.
After the test statistic is computed, this information can be used to determine
whether the null hypothesis should be rejected or accepted, just like is done
with a z test.
The p-value for the test statistic can be obtained using statistical software.
A table can also be used to decide whether to accept or reject the null
hypothesis.
Choose a sample of size n from a large population that contains an unknown
mean µ. To test the hypothesis H
0
: µ = µ
0
, compute the one-sample t statistic:
Find the P-value by calculating the probability of getting a t statistic this large or
larger in the direction specified by the alternative hypothesis H
a
in a t-
distribution with df = n – 1.