2024 ACJC H1 Q12

In this question you should state the parameters of any normal distributions you use.

In a certain college, the heights of boys and girls have independent normal distributions. The heights of boys are normally distributed with mean \(172\) cm and standard deviation \(\sigma\) cm. The probability that the boys are taller than \(156\) cm is \(0.948\).

1. Show that \(\sigma = 9.84\), correct to \(3\) significant figures.[2]

Use \(\sigma = 9.84\) for the rest of this question.

2. A sample of \(100\) randomly chosen boys are taken. \(\overline{X}\) denote the mean height of the boys in the sample. \(80\) such samples are taken.
   1. Find the expected number of samples where \(\overline{X} > 172\).[2]
   2. Find the probability that at most \(35\) samples have \(\overline{X} > 172\).[2]

The heights of girls are normally distributed with mean \(165\) cm and standard deviation \(8\) cm.

3. A boy and a girl are chosen at random. Find the probability that their heights differ by less than \(5\) cm.[3]
4. Find the probability that the total height of \(3\) randomly chosen girls exceeds twice the height of a randomly chosen boy by at most \(149\) cm.[4]