2024 ASRJC H1 Q12

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

In a botanical study, the masses of various types of plants are analysed. The masses, in grams, of Type A plants and Type B plants can be modelled by independent normal distributions. The means and standard deviations of these distributions are shown in the following table.

1. It is \(95\)% certain that \(6\) Type A plants chosen at random have a total mass exceeding \(L\) grams. Find the value of \(L\), correct to \(1\) decimal place.[2]
2. Find the probability that the total mass of two randomly chosen Type A plants is within \(40\) grams of twice the mean mass of the Type B plants.[3]

A specific experiment requires the use of \(6\) Type A plants and \(8\) Type B plants. The experiment preparation process reduces the mass of each Type A plant by \(7\)% and the mass of each Type B plant by \(5\)%.

3. Find the probability that the total mass, after preparation, of \(6\) randomly chosen Type A plants and \(8\) randomly chosen Type B plants is at least \(2.4\) kilograms.[3]

Type A plants are sold at \(\$20\) per \(100\) grams, and Type B plants are sold at \(\$17.50\) per \(100\) grams in a plant nursery. Let \(W\) represent the mass of a randomly chosen Type A plant, and \(V\) represent the mass of a randomly chosen Type B plant.

4. Find \(\mathrm{P}(3(0.2W) - 0.175(V_1 + V_2) > 57)\) and explain in the context of this question what your answer represents.[4]