In a food processing plant, two different machines are used to produce packets of flour. Over a long period of time it has been established that the masses, in \(\text{kg}\), of packets produced by Machine \(P\) follow the distribution \(\mathrm{N}(2.2, 0.1^2)\) and the masses, in \(\text{kg}\), of packets produced by Machine \(Q\) follow the distribution \(\mathrm{N}(2.1, 0.05^2)\). It is assumed that these two distributions are independent.
Explain why, in the context of the question, it is reasonable to assume that the two distributions are independent.[1]
Find the probability that a randomly chosen packet produced by Machine \(P\) has a greater mass than a randomly chosen packet produced by Machine \(Q\).[3]
Find the probability that \(3\) randomly chosen packets produced by Machine \(P\) and \(5\) randomly chosen packets produced by Machine \(Q\) have a total mass greater than \(17\text{ kg}\).[3]
Following an adjustment to Machine \(P\), the production manager wishes to test if the mean mass of packets produced by that machine now differs from \(2.2\text{ kg}\).
State hypotheses for the production manager's test, defining any parameter that you use.[2]
The production manager finds the masses of a random sample of \(30\) packets produced by Machine \(P\). The masses, \(x\text{ kg}\), are summarised below.
\(\sum (x - 2) = 4.5 \qquad \sum (x - 2)^2 = 1.11\)
Calculate unbiased estimates of the population mean and variance of the masses of packets after Machine \(P\) has been adjusted.[2]
Carry out the production manager's test at the \(5\%\) level of significance. Show the values you use to carry out the test and give your conclusion in the context of the question.[4]
Explain why the production manager would have less confidence in the conclusion of the hypothesis test if a sample of fewer than \(30\) packets had been used.[1]
Explain why the production manager would have less confidence in the conclusion of the hypothesis test if the sample had not been chosen randomly.[1]