A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s _ { A } ^ { 2 } = 0.495 \mathrm {~mm} ^ { 2 }\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s _ { B } ^ { 2 } = 1.04 \mathrm {~mm} ^ { 2 }\).
Stating your hypotheses clearly test, at the \(10 \%\) significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\).
State the assumption you have made about the populations of pebble lengths in order to carry out the test.
A random sample of 10 mustard plants had the following heights, in mm , after 4 days growth.
$$5.0,4.5,4.8,5.2,4.3,5.1,5.2,4.9,5.1,5.0$$
Those grown previously had a mean height of 5.1 mm after 4 days. Using a \(2.5 \%\) significance level, test whether or not the mean height of these plants is less than that of those grown previously.
(You may assume that the height of mustard plants after 4 days follows a normal distribution.)