5. In a large population of hens, the weight of a hen is normally distributed with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 hens is taken from the population. The mean weight for the sample is denoted \(\bar { X }\).
a. State the distribution of \(\bar { X }\) giving its mean and variance. The sample values are summarised by \(\sum x = 199.8\) and \(\sum x ^ { 2 } = 407.8\) where \(x \mathrm {~kg}\) is the weight of a hen.
b. Find an unbiased estimate for \(\mu\).
c. Find an unbiased estimate for \(\sigma ^ { 2 }\).
d. Find a \(90 \%\) confidence interval for \(\mu\). It is found that \(\sigma = 0.27\). It is decided to test, at the \(1 \%\) level of significance, the null hypothesis \(\mu = 1.95\) against the alternative hypothesis \(\mu > 1.95\).
e. Find the \(p\)-value for the test.
f. Write down the conclusion reached.
g. Explain whether or not the central limit theorem was required in part e.