Digital watches of a certain brand are advertised to have an average lifetime of 800 days. A customer is willing to purchase a watch unless it can be demonstrated that the average lifetime is shorter than what is advertised. A random sample of 50 watches is obtained from the manufacturer and the lifetime of each watch is recorded. In particular, one obtains the sample mean
X
ˉ
=738.44
days and sample variance
S 2
=1459.24
days squared from this sample of digital watches. 1. Explain why the test statistic
ζ= S/ 50
X
ˉ
−μ
has a normal distribution despite the experimenter being unaware of the distribution of the population of digital watches. 2. Devise an appropriate hypothesis test to determine whether the given customer is willing to purchase a digital watch. 3. Conduct the hypothesis test of 2 . at a significance level of
α=0.05
