A team of engineers tests the operational lifespan of a new emergency beacon under sub-zero temperatures. A random sample of 12 beacons is activated, and the time until failure, ttt hours, is recorded. The data are summarized as follows:
∑t=504,∑t2=21450 \sum t = 504, \quad \sum t^2 = 21450 ∑t=504,∑t2=21450You may assume that the lifespans are normally distributed.
Calculate a 98% confidence interval for: (i) the mean lifespan of the beacons, (ii) the variance of the lifespan of the beacons.
Beacons that fail in less than 40 hours are designated as "short-life". Using the relevant confidence limits from part (a), determine the lowest estimate for the proportion of beacons that are short-life.