N2024 P2 Q10

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

A company makes metal plates that can be used to fix fence panels onto posts. The metal plates have four small holes drilled into them for screws and two large holes drilled into them for bolts (see diagram).

Before the holes are drilled, the masses of plates, in grams, follow the distribution \(\mathrm{N}(200, 1.6^2)\).

1. Find the probability that the mass of a randomly chosen plate before drilling is more than \(197.5\) grams.

   [1]

Drilling the holes reduces the mass of each plate by \(5\%\). A production worker selects 8 of the drilled plates at random.

2. Find the probability that at least \(5\) of these \(8\) plates have masses between \(190\) grams and \(192\) grams.

   [4]

The drilled plates are sold in packs of \(20\) randomly chosen plates.

3. Find the probability that the total mass of a pack of \(20\) drilled plates is less than \(3805\) grams.

   [3]

The manufacturer decides to sell ‘Value’ packs containing all the materials needed to fix fence panels onto posts. Each Value pack consists of \(20\) drilled plates together with the right number of screws and bolts to fit them; all of these are randomly chosen.

• The masses of screws, in grams, follow the distribution \(\mathrm{N}(10, 0.3^2)\).
• The masses of bolts, in grams, follow the distribution \(\mathrm{N}(44, 1.1^2)\).

4. Find the mass exceeded by just \(5\%\) of the Value packs. Give your answer to the nearest gram. You should ignore the mass of any packaging.

   [4]