Hypothesis Testing for the Binomial Distribution

Question

Consider the following problem:

$H_0:p=0.2, H_1: p\neq 0.2$

$X\sim B(25,0.2)$

Find the critical region for a hypothesis test using a $5\%$ significance level.

I have found

$$\begin{aligned} P(X=0)&=0.0038\\ P(X\leq1)&=0.0274\\ P(X\geq9)&=0.0468\\ P(X\geq10)&=0.0173\\ \end{aligned}$$

I therefore said that the critical region is $X=0$ and $X\geq10$, because we require the probability in each tail to be at most $0.025$.

However, the textbook answers say that the critical region is $X\leq1$ and $X\geq10$. They say that $1$ should be selected rather than $0$ because $0.0274$ is closer to $0.025$ than $0.0038$.

Which of these is correct? Or are they both correct?

Answer 1

The answer depends on the definition of the rejection region. Here four possibilities:

The rejection region is given by $[0,c_1]\cup [c_2,25]$ such that

1. $P(X\leq c_1)\leq \frac{\alpha}{2}$ and $P(X\geq c_2)\leq\frac{\alpha}{2}$
2. $P(X\leq c_1)+P(X\geq c_2)\leq \alpha$
3. $P(X\leq c_1)\approx \frac{\alpha}{2}$ and $P(X\geq c_2)\approx\frac{\alpha}{2}$
4. $P(X\leq c_1)+P(X\geq c_2)\approx \alpha$

You see, that definition 3 and 4 are a little bit more handwaving.

For definition 1, you have to take $c_1=0$ and $c_2=10$. But with definiton 2 you can choose $c_1=1$ and $c_2=10$.