Confidence interval supports assertion

5. A machine is filling bottles of milk. A random sample of 16 bottles was taken and the volume of milk in each bottle was measured and recorded. The volume of milk in a bottle is normally distributed and the unbiased estimate of the variance, \(s ^ { 2 }\), of the volume of milk in a bottle is 0.003

1. Find a 95\% confidence interval for the variance of the population of volumes of milk from which the sample was taken. The machine should fill bottles so that the standard deviation of the volumes is equal to 0.07
2. Comment on this with reference to your 95\% confidence interval.

2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$

1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)