2024 ACJC H1 Q7

It is known that on average \(p\)% of the elderly patients in ABC hospital suffer from hypertension. A research study on \(30\) randomly selected elderly patients in the hospital was conducted. The number of elderly patients in the hospital with hypertension is denoted by the random variable \(X\). Assume that \(X\) has a binomial distribution.

1. If \(X\) has mean \(18\), calculate \(P(X \ge 20)\).[3]

It is also known that on average \(k\)% of the elderly patients in ABC hospital suffer from diabetes. The hospital conducted a research study on another \(40\) randomly selected elderly patients. The number of elderly patients in the hospital with diabetes also follows a binomial distribution.

2. The probability of at least nine but no more than twenty elderly patients suffering from diabetes is \(0.25\). Given that \(k < 50\), find the value of \(k\).[3]