Power of a serve test

Topic: Hypothesis tests · more questions on this topic
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A player’s first-serve successes are modelled as X\sim\operatorname{Bin}(15,p). The test of H_0:p=0.6 against H_1:p>0.6 rejects when X\in\{13,14,15\}.

1. Compute the size of the test, that is the power at p=0.6.

2. Write the power as a function of p, and evaluate it at p=0.9.

Worked answer (click to show)

Under H_0, X\sim\operatorname{Bin}(15,0.6) and

\mathbb{P}(X\ge 13)=\sum_{k=13}^{15}\binom{15}{k}(0.6)^k(0.4)^{15-k}\approx 0.0271.

That is the size of the test. At a general p,

\mathrm{Power}(p)=\sum_{k=13}^{15}\binom{15}{k}p^k(1-p)^{15-k}.

At p=0.9 the power is about 0.816. At p=0.8 it is only 0.398: with 15 serves, a real improvement from 60% to 80% would be missed more often than not.