The apple trees in a large orchard have, for several years, suffered from a disease for which the outward sign is a red discolouration on some leaves.

The fruit grower knows that the mean number of discoloured leaves per tree is 42.3. The fruit grower suspects that the disease is caused by an infection from a nearby group of cedar trees. He cuts down the cedar trees and, the following year, counts the number of discoloured leaves on a random sample of seven apple trees. The results are given in the table below.

(a)     From these data calculate an unbiased estimate of the population variance.

(b)     Stating null and alternative hypotheses, carry out an appropriate test at the 10 % level to justify the cutting down of the cedar trees.

(a)     \(n = 7,{\text{ sample mean }} = 35\)     (A1)

\(s_{n - 1}^2 = \frac{{\sum {{{(x - 35)}^2}} }}{6} = 322\)     (M1)A1

[3 marks]

(b)     null hypothesis \({{\text{H}}_0}:\mu = 42.3\)     A1

alternative hypothesis \({{\text{H}}_1}:\mu < 42.3\)     A1

using one-sided t-test

\(\left| {{t_{{\text{calc}}}}} \right| = \sqrt 7 \frac{{42.3 - 35}}{{\sqrt {322} }} = 1.076\)     (M1)(A1)

with 6 degrees of freedom , \({t_{{\text{crit}}}} = 1.440 > 1.076\)

\({\text{(or }}p{\text{-value }} = 0.162 > 0.1)\)     A1

we conclude that there is no justification for cutting down the cedar trees     R1     N0

Note: FT on their t or p-value.

[6 marks]

Total [9 marks]

This question was generally well attempted as an example of the t-test. Very few used the Z statistic, and many found p-values.