2023 RVHS P2 Q11

Rectangular blocks of wood undergo crafting and lacquering to become ornaments. The masses of the rectangular blocks, \(X\) g, are known to have normal distribution with mean \(300\)g and standard deviation \(20\) g. Crafting the company logo on a block reduces its mass by \(10\%\). After that, a layer of lacquer is applied to the crafted wooden block that increases its mass by \(5\%\).

1. Show that the probability that a piece of completed ornament has mass between \(290\) g and \(350\) g is approximately \(0.365\).[3]
2. During an inspection, completed ornaments are selected randomly and their masses are recorded. Find the probability that the \({{10}^{\mathrm{th}}}\) ornament selected is the \({{3}^{\mathrm{rd}}}\) piece that has mass between \(290\) g and \(350\)g.[2]

The ornament is placed into a gift box before it is sold. The masses of the gift boxes are modelled as \(100\alpha \%\) of the masses of the ornaments.

3. Given that \(40\%\) of the total mass of an ornament and its box exceeds \(360\) g, determine the value of \(\alpha \).[5]
   In another batch of wooden blocks, the manufacturer claims that the masses of the block have normal distribution with mean \(130\)g and standard deviation \(80\)g. Justify if this distribution is appropriate.[2]