Problem

Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, $X$ cm, of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm.

She collects a random sample of 80 pine cones and their lengths are recorded in the table below.

Length, x cm	$x<5$	$5 \leq x < 5.5$	$5.5 \leq x < 6$	$6 \leq x < 6.5$	$x \geq 6.5$
Frequency	6	14	24	26	10

(a) Stating your hypotheses clearly and using a 10% level of significance, test Chrystal’s belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.

Chrystal’s friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that

$$\Sigma x = 464 \qquad \text{and} \qquad \Sigma x^2 = 2722.59$$

(b) Calculate unbiased estimates of the mean and variance of the lengths of the pine cones.

David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal’s sample.

His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.

(c) Using a 10% level of significance, complete David’s test stating the critical value and the degrees of freedom used.

(d) Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm.