Independent samples t-test - computational notes
In the Independent samples t-test, the difference between the observed means in two independent samples is calculated. A significance value (P-value) and 95% Confidence Interval (CI) of the difference is reported. The P-value is the probability of obtaining the observed difference between the samples if the null hypothesis were true. The null hypothesis is the hypothesis that the difference is 0.
The program first calculates the pooled standard deviation s:
$$ s = \sqrt { \frac { (n_1-1) s_1^2 + (n_2-1) s_2^2 } { n_1+n_2-2 } } $$
$$ s = \sqrt { \frac { (n_1-1) s_1^2 + (n_2-1) s_2^2 } { n_1+n_2-2 } } $$ where s1 and s2 are the standard deviations of the two samples with sample sizes n1 and n2.
The standard error se of the difference between the two means is calculated as:
$$ se(\bar{x_1} - \bar{x_2}) = s \times \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } $$
$$ se(\bar{x_1} - \bar{x_2}) = s \times \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } $$ The significance level, or P-value, is calculated using the t-test, with the value t calculated as:
$$ t = \frac {\bar{x_1} - \bar{x_2}} { se(\bar{x_1} - \bar{x_2}) } $$
$$ t = \frac {\bar{x_1} - \bar{x_2}} { se(\bar{x_1} - \bar{x_2}) } $$