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

Question

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)?

Answer 1

I hope this answers your questions:

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.)

2. 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.