# 17.3.3.2 Algorithms (Pair-Sample T-Test)

## Contents

This function is used to test whether or not the difference of the two paired sample means equals to $\mu_d\,\!$(i.e. to test whether or not their means are equal, you can just test whether or not their difference is 0, $\mu_1-\mu_2=\mu_d=0\,\!$ ). And the hypotheses take the form: $H_0:\mu_1-\mu_2=\mu_d\,\!$ vs $H_1:\mu_1-\mu_2 \ne \mu_d$ Two Tailed $H_0:\mu_1-\mu_2 \le \mu_d$ vs $H_1:\mu_1-\mu_2 > \mu_d$ Upper Tailed $H_0:\mu_1-\mu_2 \ge \mu_d$ vs $H_1:\mu_1-\mu_2 < \mu_d$ Lower Tailed

### Test Statistics

Consider two samples $x_1\,\!$and $x_2\,\!$ which assumed to be drawn from normal populations are equal size, we can define the paired difference as: $d_j=x_{1j}-x_{2j},for(j=1,2,...,n)\,\!$

And we have the mean paired difference is: $\bar{d}=\frac{1}{n}\sum_{i=1}^n d_i$

Then we can compute the standard deviation for the difference between paired data points $s_d\,\!$, with v = n-1 degrees of freedom as: $s_d=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(d_i-\bar{d})}$

And then we can calculate the test statistic by: $t=\frac{\bar{d}-\mu_d}{\frac{s_d}{\sqrt{n}}}$

Compare the t value with the critical value and we will reject $H_0\,\!$ if:

Two tailed test: $|t| > t_{\sigma/2}\,\!$;

Upper tailed test: $t > t_\sigma\,\!$;

Lower tailed test: $t < -t_\sigma\,\!$;

The p-value will also be compared with a user-defined significance level,, $\sigma\,\!$, which commonly 0.05 is used. And the null hypothesis $H_0\,\!$ will be rejected if $p < \sigma\,\!$.

### Confidence Intervals

The confidence interval for paired sample mean difference $(\mu_1 - \mu_2)\,\!$ is:

Null Hypothesis Confidence Interval $H_0:\mu_1-\mu_2=\mu_d\,\!$ $\left[\bar{d} - t_{\alpha/2}\frac{s_d}{\sqrt{n}}, \bar{d} + t_{\alpha/2}\frac{s_d}{\sqrt{n}}\right]$ $H_0:\mu_1-\mu_2 \le \mu_d$ $\left[\bar{d} - t_{\alpha}\frac{s_d}{\sqrt{n}}, \infty\right]$ $H_0:\mu_1-\mu_2 \ge \mu_d$ $\left[-\infty, \bar{d} + t_{\alpha}\frac{s_d}{\sqrt{n}}\right]$

### Power Analysis

The power of a two sample t-test is a measurement of its sensitivity. Detail algorithm about calculating power please read the help of Power and Sample Size.