Linear Combinations, Statistical Modeling, and Probabilistic Inference

IB AA SL and HL: core and optional content

Core: the single-boy weight percentile and one girl/one boy on opposite sides of 47 kg. Optional: the difference of two girls’ weights and the pooled mean weight.

The weights of girls and boys in a college have independent normal distributions with their means and standard deviations given in the following table.

1. Find the greatest weight (correct to nearest \(0.1\) kg) exceeded by at least \(10\%\) of the boys. [2]
2. Find the probability that the difference of the weights of \(2\) randomly chosen girls is less than \(1.2\) kg. [2]
3. One girl and one boy are chosen at random. Find the probability that one of them is at least \(47\) kg and the other is less than \(47\) kg. [3]
4. Find the probability that the average of \(2\) randomly chosen girls and \(8\) randomly chosen boys is less than \(50\) kg.[3]

Modelling assumption for this exercise: weights of distinct selected students are mutually independent and follow the stated distribution for their group.