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.

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 t⁹_0.025 at the intersection of the row representing 9 degrees in the Degrees of Freedom column and the t_0.025 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.

Critical value taken from the t-distribution: two-tailed test.

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.