Discussion
Based on a sample of size one from Binomial (6, P), it is decided to test the hypothesis \(H_{o} : P = 1/2\) Vs \(H_{i} : P = 1/4\) using the critical region \(X \le 3\). Then, the size of the critical region would be :
Answer: D
The size is 42/64.
The size of a test is the probability of the critical region when \(H_{0}\) is true, here \(P = \frac{1}{2}\) with n = 6:
\[ \begin{aligned} P(X \le 3) &= \frac{\binom{6}{0} + \binom{6}{1} + \binom{6}{2} + \binom{6}{3}}{2^{6}} \\ &= \frac{1 + 6 + 15 + 20}{64} = \frac{42}{64} \end{aligned} \]
22/64 stops at \(X \le 2\), 35/64 and 15/64 do not correspond to the region \(X \le 3\) under \(P = \frac{1}{2}\). The alternative \(P = \frac{1}{4}\) would be used for the power, not the size.
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