Find the Critical Two-Tailed Values When Testing a Hypothesis for a Small Sample
When you use a small sample to test a hypothesis about a population mean, you take the resulting critical value or values from the Student’s t-distribution. For a two-tailed test, the critical value is
and n represents the sample size.
|Degrees of Freedom||t0.10||t0.05||t0.025||t0.01||t0.005|
The number of degrees of freedom used with the t-distribution depends on the particular application. For testing hypotheses about the population mean, the appropriate number of degrees of freedom is one less than the sample size (that is, n – 1).
The critical value or values are used to locate the areas under the curve of a distribution that are too extreme to be consistent with the null hypothesis. For a two-tailed test, the value of the level of significance
is split in half; the area in the right tail equals
and the area in the left tail equals
As an example of a two-tailed test, suppose the level of significance is 0.05 and the sample size is 10; then you get a positive and a negative critical value:
You can get the value of the positive critical value
directly from the Student’s t-distribution table.
In this case, you find the positive critical value t90.025 at the intersection of the row representing 9 degrees in the Degrees of Freedom column and the t0.025 column. The positive critical value is 2.262; therefore, the negative critical value is –2.262. You represent these two values like so:
You represent them graphically as shown here.
The shaded region in the two tails represents the rejection region; if the test statistic falls in either tail, the null hypothesis will be rejected.