Type II error of a coin test when heads has chance 0.7

Question

A coin is tossed $36$ times and fairness is rejected when the head count $X$ satisfies $|X-18|\ge4$ . If the coin actually lands heads with probability $0.7$ , what is the type II error probability $\beta$ , to four decimal places?

Short answer

About $0.0916$ . We fail to reject when $15\le X\le21$ . With $X\sim\text{Bin}(36,0.7)$ , $\mathbb{P}(X\le21)-\mathbb{P}(X\le14)\approx0.0916$ , because $21$ lies well below the mean of $25.2$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Populations, Samples, and Sampling Distributions
Sample Spaces, Events, and the Probability Axioms

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