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For a binomial distribution, the hypothesis $H_0: p = 0.2$ and $H_1: p\ne0.2$ are tested at the $10%$ level. $20$ trials are performed and the critical region is $X = 0$ or $X > 7$. Calculate the probability of a type II error and the power when the true value of $p$ is $0.3$:

My workings: $X - B(20, 0.3)$

The probability of a type II error is the probability of getting a result outside of the critical region when $p = 0.3$.

Hence $P(\text{Type II error}) = P(1 \le X \le 7)$

Hence $P = 0.771$

The textbook says $P = 0.772$, which is the $P X \ge 7$

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I think your working is fine.

The book did indeed provided the value of $P(X \le 7)$ and it seems that they forget to subtract $P(X=0)$.

> pbinom(7,20, 0.3)
[1] 0.7722718
> pbinom(7,20, 0.3)-pbinom(0,20,0.3)
[1] 0.7714739
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