6. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with mean \(\mu\).
- Show that \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\) is an unbiased estimator of the population mean \(\mu\).
A company produces small jars of coffee.
Five jars of coffee were taken at random and weighed.
The weights, in grams, were as follows
$$\begin{array} { l l l l l }
197 & 203 & 205 & 201 & 195
\end{array}$$
- Calculate unbiased estimates of the population mean and variance of the weights of the jars produced by the company.
It is known from previous results that the weights are normally distributed with standard deviation 4.8 g .
The manager is going to take a second random sample. He wishes to ensure that there is at least a \(95 \%\) probability that the estimate of the population mean is within 1.25 g of its true value.
- Find the minimum sample size required.