# Interpretation of the region of rejection in hypothesis testing in binomial distribution

The pharmacy company Life Co. has developed a new drug against insomnia. To check the effectiveness, this drug was tested with n = 10 patients. At present, the standard medication can cure 30% of the treated patients.

• The treatment with the new drug was successful with exactly four patients. Perform a one-sided hypothesis test to decide if the new drug is better than the standard one (with a level of significance of 1%). Write down explicitly all six steps.:
• Model: $$X$$ is the number of patients which were succesffully treated, $$X\sim \text{Bin}(10, \pi)$$

• Null hypothesis: $$H_0 : \pi = 0.3$$. Alternative hypothesis: $$H_A: \pi > 0.3$$

• Test statistic: $$X$$ - number of cured patients Distribution under $$H_0 : X\sim \text{Bin}(10, 0.3)$$
• Choose significance level: $$\alpha = 1\% = 0.01$$
• Range of rejection (note: one-sided test): We look for set $$K = \{...\}$$ such that $$P_{H_0}(X\in K)\leq \alpha$$

$$\begin{array}{l|llllll} &x = 5&x = 6&x = 7&x = 8&x = 9&x = 10\\ \hline P(X \geq x)&0.1503&0.0473&0.0106&0.0016&0.0001&5.9\cdot 10^{-6}\\ \end{array}$$ Therefore the rejection range is $$K = {8,9,10}$$. The probabilities listed in the table can be calculated in R in the following way

    # R Code
> n=10
> pi=0.3
> 1-pbinom(4:9,n,pi)
[1] 0.1502683326 0.0473489874 0.0105920784 0.0015903864
[5] 0.0001436859 0.0000059049

• Test decision: Since $$4 \not \in K$$, $$H_0$$ cannot be rejected. Therefore, we cannot proof that the success rate of the new drug is better

HERE ARE MY QUESTIONS:

1. How do we find the number of the rejection range $$K = 8,9,10$$
2. Where does the number 4:9 come from in the code 1-pbinom(4:9,n,pi)?

1. Since your hypothesis test is being performed at the $$0.01$$ significance level, $$K = 8,9,10$$ is the set of $$x$$ values for which you can reject $$H_{0}$$ at that significance level. In other words, those three $$x$$ values produce a $$p$$-value that is lower than your pre-determined significance level. You can see that those three $$p$$-values are all smaller than $$0.01$$. For all other values of $$x$$, $$p$$-values are larger than $$0.01$$.
(This is just another way of performing a hypothesis test. You can calculate a $$p$$-value for the value of $$x$$ in your alternative hypothesis (4 in you case) and then see if it is smaller than $$\alpha$$. Or you can do what was done here and first determine values of $$x$$ that allow you to reject $$H_{0}$$, and then see if the value you tested is one of those values.)
1. 1-pbinom(4,n,pi) means $$1 - P(X \leq 4)$$ or, equivalently, $$P(X \geq 5)$$. So, 1-pbinom(4:9,n,pi) will produce, separately, $$P(X \geq 5)$$ through $$P(X \geq 10)$$. This code is provided to illustrate how that table of $$p$$-values right above the code can be generated, based on the specific parameters of your Binomial distribution.