7. The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows.
$$\begin{array} { l l l l l l l l l l }
498 & 502 & 500 & 496 & 509 & 504 & 511 & 497 & 506 & 499
\end{array}$$
- Find unbiased estimates of the mean and the variance of the population from which this sample was taken.
Given that the population standard deviation is 5.0 g ,
- estimate limits, to 2 decimal places, between which \(90 \%\) of the weights of the tubs lie,
- find a \(95 \%\) confidence interval for the mean weight of the tubs.
A second random sample of 15 tubs was found to have a mean weight of 501.9 g .
- Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean weight of these tubs is greater than 500 g .