Use sample variance and the appropriate t degrees of freedom.
Notes and key methods
For independent observations from a normal population of unknown variance, \(\displaystyle t= \frac{\bar{x}- \mu_{0}}{\frac{s}{\sqrt{n}}}\) has n−1 degrees of freedom under \(\displaystyle H_{0}\). The sample variance \(\displaystyle s^{2}\) uses denominator n−1.
A confidence interval uses \(\displaystyle \bar{x}\pm \frac{t_{n- 1, 1- \frac{\alpha }{2}}s}{\sqrt{n}}\). The t distribution allows uncertainty from estimating σ; replacing its quantile by z can give an interval that is too narrow for a small sample.
Check the conditions
A normal population is assumed for exact small-sample t inference.
The observations are a random independent sample.
Three worked examples
Example 1 · One-sample t inference
A random sample of 8 independent normal observations has mean 24 and unbiased sample standard deviation 4. Test \(H_0:\mu=22\) against an increase; the population variance is unknown.
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the observed t statistic.
Find the degrees of freedom.
Find the one-sided p-value.
Find the half-width of a two-sided 95% t confidence interval.
The p-value is \(0.10010003708312\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population mean exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Example 2 · One-sample t inference
A random sample of 9 independent normal observations has mean 25.5 and unbiased sample standard deviation 5. Test \(H_0:\mu=23\) against an increase; the population variance is unknown.
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the observed t statistic.
Find the degrees of freedom.
Find the one-sided p-value.
Find the half-width of a two-sided 95% t confidence interval.
The p-value is \(0.0860016459759556\). Since it exceeds 0.05, do not reject \(\displaystyle H_{0}\): the sample gives insufficient evidence that the population mean exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Example 3 · One-sample t inference
A random sample of 10 independent normal observations has mean 27 and unbiased sample standard deviation 2. Test \(H_0:\mu=24\) against an increase; the population variance is unknown.
Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.
Find the observed t statistic.
Find the degrees of freedom.
Find the one-sided p-value.
Find the half-width of a two-sided 95% t confidence interval.
The p-value is \(0.000526935628508277\). Since it is at most 0.05, reject \(\displaystyle H_{0}\): there is evidence that the population mean exceeds its null value. This does not give the probability that \(\displaystyle H_{0}\) is true.
Common mistakes
Use n−1 degrees of freedom, not n.
Keep the one-sided test tail distinct from a two-sided interval.