a) The nails produced by a machine are assumed to have a mean of 5.0 cm. A random sample of 80 nails gives a mean of 5.2 cm and a standard deviation of 1.3 cm. Is there evidence that the population mean has increased at the 5% level? (10 marks) b) In a butter-packing plant, the quantity of butter packed in a day using a certain type of machine is normally distributed with mean / and variance 4. On a particular day, 12 packets of butter were taken at random from this production line and their masses, measured in grams, were: 9.5, 9.5, 11.2, 10.6, 9.9, 11.1, 10.9, 9.8, 10.1, 10.2, 10.9, 11.0 i. Find the 95% confidence interval for the mean mass produced by this machine. (5 marks) ii. At 1% level of significance, estimate the mean mass produced by this machine. (3 marks) Based on your answer in (1) and (ii), what can you observe when the confidence level is increased? (1 mark)