The random variable \(X\), which has distribution \(\mathrm { N } \left( \mu _ { X } , \sigma ^ { 2 } \right)\), is independent of the random variable \(Y\), which has distribution \(\mathrm { N } \left( \mu _ { Y } , \sigma ^ { 2 } \right)\).
In order to test \(\mathrm { H } _ { 0 } : \mu _ { X } = 1.5 \mu _ { Y }\), samples of size \(n\) are taken on each of \(X\) and \(Y\) and the random variable \(\bar { D }\) is defined as
$$\bar { D } = \bar { X } - 1.5 \bar { Y }$$
State the distribution of \(\bar { D }\) assuming that \(\mathrm { H } _ { 0 }\) is true.
A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams.
The machine fills two sizes of bags: large and extra-large.
The mean weight of rice in a random sample of 50 large bags is 1509 grams.
The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams.
Test, at the \(5 \%\) level of significance, the claim that, on average, the rice in an extra-large bag is \(1 \frac { 1 } { 2 }\) times as heavy as that in a large bag. [0pt]
[6 marks]