1 A machine is set to fill pots with yoghurt such that the mean weight of yoghurt in a pot is 505 grams.
To check that the machine is working properly, a random sample of 8 pots is selected. The weight of yoghurt, in grams, in each pot is as follows
$$\begin{array} { l l l l l l l l }
508 & 510 & 500 & 500 & 498 & 503 & 508 & 505
\end{array}$$
Given that the weights of the yoghurt delivered by the machine follow a normal distribution with standard deviation 5.4 grams,
- find a \(95 \%\) confidence interval for the mean weight, \(\mu\) grams, of yoghurt in a pot. Give your answers to 2 decimal places.
- Comment on whether or not the machine is working properly, giving a reason for your answer.
- State the probability that a \(95 \%\) confidence interval for \(\mu\) will not contain \(\mu\) grams.
- Without carrying out any further calculations, explain the changes, if any, that would need to be made in calculating the confidence interval in part (a) if the standard deviation was unknown. Give a reason for your answer.
You may assume that the weights of the yoghurt delivered by the machine still follow a normal distribution.