Two-sample t-Tests

Two-sample t-test under scenario 1

This is often referred to as the two-sample pooled t-test. If we have two independent samples from two normal distributions with equal variances ($\sigma^{2}_{1960}$ = $\sigma^{2}_{1990}$ = $\sigma^2$ ), the test statistic for the null hypothesis is as follows:

$$t = \frac{(\bar{y}_{1960} - \bar{y}_{1990}) - (\mu_{1960} - \mu_{1990})}{{\sqrt[sp]{\frac{1}{n_{1960}} + \frac{1}{n_{1990}}}}}$$

Above, $s_p$ is the pooled sample standard deviation, computed as follows:

$$s_p = \frac{(n_{1960} - 1)s^2 _{1960} + (n_{1990} - 1)s^2 _{1990}}{n_{1960} + n_{1990} - 2}$$

The test statistic follows a t-distribution, with a degree of freedom ($n_{1960}$ + $n_{1990}$ − 2). So, the computed t-statistic, together with the degree of freedom, will allow us to identify the $p$ value from the t probability distribution table. If the $p$ value is larger than 0.05 or 5%, the null hypothesis won’t be rejected. If the $p$ value is smaller than 0.05 or 5%, the null hypothesis will be rejected. The R code for the test is as follows: