A metal cutter produces metal cuts from a running metal band. The pieces must have a certain length μ, but the cutter is somewhat unreliable. Even when it is set to cut the desired length, fluctuations in the length of the metal band may occur. Assume that the length has a normal distribution.

(a) You take a sample of 17 cuts from the metal band and determine an average value of 150 cm with an empirical variance of S² = 5.76 cm². Determine the 80% confidence interval for the average width of the metal band.

(b) What is the smallest possible sample size so that a 90% two-sided confidence interval of the average length of the metal pieces still has a length of 1.2 cm?

(c) What is the smallest variance that still permits us to assume a 90% confidence interval?