4 A random sample of 160 observations of a random variable \(X\) is selected. The sample can be summarised as follows.
\(n = 160 \quad \sum x = 2688 \quad \sum x ^ { 2 } = 48398\)
- Calculate unbiased estimates of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- Find a 99\% confidence interval for \(\mathrm { E } ( X )\), giving the end-points of the interval correct to 4 significant figures.
- Explain whether it was necessary to use the Central Limit Theorem in answering
- part (a),
- part (b).