17.9.2.2 Algorithms ((PSS) Two-Sample T-Test)

For two equal size (n) samples, the Power with degrees of freedom \(v = 2(n-1)\) and pooled standard deviation \(s\) is calculated by:

• For a Greater than Alternative Hypothesis:

\[ Power=Prob(t >t_{(1-\alpha ,\nu )},\nu ,\lambda )\,\!\]

• For a Less than Alternative Hypothesis:

\[ Power=Prob(t <t_{(\alpha ,\nu )},\nu ,\lambda )\,\!\]

• For a Not Equal Alternative Hypothesis:

\[ Power=Prob(t >t_{(1-\alpha /2,\nu )},\nu ,\lambda )+ Prob(t <t_{(\alpha /2,\nu )},\nu ,\lambda )\,\!\]

Where the noncentrality parameter:

\[\lambda =\delta \sqrt{n} \,\!\]

and

\[ \delta =\frac{ \mu _\alpha -\mu _0 }s\,\!\]