Let's say ABC drug company claims their medicine cures 90% of patients. Now a doctor conducts a trial by taking a random sample of 15 patients. Result shows 11 of them are cured but 4 are not.
In this case, $H_0$ should be: $p=0.9$. And $H_1$ is: $p<0.9$. If we use $X$ to represent the number of people cured in the sample, then $X \sim B(15,0.9)$.
If significance level ($\alpha$) is set as 5%, the critical region in this case should be defined as:
$P(X<11)<\alpha$ where $\alpha=5\%$ , $H_0$ will be rejected if the expression is true.
This really confuses me. The smaller of $P(X<11)$ is, the larger of $P(X>11)$ should be. Then the more patients should be cured and more likely H0 should be accepted. Anything wrong with this?
To me, it seems to make more sense if $P(X>11) < \alpha$ proves to be true to reject $H_0$.