N2009 P2 Q9

The thickness in cm of a mechanics textbook is a random variable with the distribution \(\mathrm N(2.5,0.1^2)\).

1. The mean thickness of \(n\) randomly chosen mechanics textbooks is denoted by \(\overline M\) cm. Given that \(\mathrm P(\overline M>2.53)=0.0668\), find the value of \(n\).[3]

The thickness in cm of a statistics textbook is a random variable with the distribution \(\mathrm N(2.0,0.08^2)\).

2. Calculate the probability that 21 mechanics textbooks and 24 statistics textbooks will fit onto a bookshelf of length 1 m. State clearly the mean and variance of any normal distribution you use in your calculation.[3]
3. Calculate the probability that the total thickness of 4 statistics textbooks is less than three times the thickness of 1 mechanics textbook. State clearly the mean and variance of any normal distribution you use in your calculation.[3]
4. State an assumption needed for your calculations in parts (ii) and (iii).[1]