Module 10: Hypothesis Testing With Two Samples
Two Population Means with Unknown Standard Deviations
Barbara Illowsky & OpenStax et al.
- The two independent samples are simple random samples from two distinct populations.
- For the two distinct populations:
- if the sample sizes are small, the distributions are important (should be normal)
- if the sample sizes are large, the distributions are not important (need not be normal)
Note: The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t-test. The degrees of freedom formula was developed by Aspin-Welch.
The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, [latex]displaystyleoverline{{X}}_{{1}}-overline{{X}}_{{2}}[/latex], and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.
Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means, [latex]displaystyleoverline{{X}}_{{1}}-overline{{X}}_{{2}}[/latex].
The standard error is: [latex]displaystylesqrt{frac{(s_1)^2}{n_1}+frac{(s_2)^2}{n_2}}[/latex]
The test statistic (t-score) is calculated as follows: [latex]frac{(overline{x}_1-overline{x}_2)-(overline{mu}_1-overline{mu}_2)}{displaystylesqrt{frac{(s_1)^2}{n_1}+frac{(s_2)^2}{n_2}}}[/latex]
Where: s1 and s2, the sample standard deviations, are estimates of σ1 and σ2, respectively. σ1 and σ1 are the unknown population standard deviations. [latex]displaystyleoverline{{x}}_{{1}}[/latex] and [latex]overline{{x}}_{{2}}[/latex] are the population means.
The number of degrees of freedom (df) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t-distribution with df as follows:
[latex]displaystyle{df}=frac{((frac{(s_1)^2}{n_1})+(frac{(s_2)^2}{n_2}))^2}{(frac{1}{n_1-1})(frac{(s_1)^2}{n_1})^2+(frac{1}{n_2-1})(frac{(s_2)^2}{n_2})^2}[/latex]When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample variances (s1)2 and (s2)2 are not pooled. (If the question comes up, do not pool the variances.)
Note: It is not necessary to compute this by hand. A calculator or computer easily computes it.
Example
The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data