Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm { N } \left( 32,4 ^ { 2 } \right)\). You may assume that the times taken to complete the crossword on successive days are independent.
(i) Find the upper quartile of \(X\) and explain its meaning in context.
(ii) Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes.
Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm { N } \left( 18,2 ^ { 2 } \right)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword.
A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows.
$$\begin{array} { l l l l l l l l l l }
68 \cdot 1 & 70 \cdot 4 & 68 \cdot 6 & 67 \cdot 7 & 71 \cdot 3 & 67 \cdot 6 & 68 \cdot 9 & 70 \cdot 2 & 68 \cdot 4 & 69 \cdot 8
\end{array}$$
You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\).
Determine a 95\% confidence interval for \(\mu\).
The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95 ?' Explain why the answer to this question is no and give a correct interpretation.