The height \(H\), in metres, of mature trees of a certain variety is normally distributed with standard deviation 0.67. In order to test whether the population mean of \(H\) is greater than 4.23, the heights of a random sample of 200 trees are measured.

(a) Write down suitable null and alternative hypotheses for the test. [1]

The sample mean height, \(\bar h\) metres, of the 200 trees is found and the test is carried out. The result of the test is to reject the null hypothesis at the 5% significance level.

(b) Find the set of possible values of \(\bar h\). [3]

(c) Ajit said, “In (b) we had to assume that H is normally distributed, so it was necessary to use the Central Limit Theorem.” Explain whether you agree with Ajit. [1]