17.9.1.2 Algorithms ((PSS) One-Sample T-Test)

For given degrees of freedom \( \nu =n-1\,\!\) and \( s\,\!\) standard deviation , the Power for one sample t-Test 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

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