17.3.3.2 Algorithms (Pair-Sample T-Test)

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.