(a) Suppose a random sample of n observations is taken from a normal population having mean mu and variance sigma^(2), so each of the observations X_(1),dots,X_(n) will have a normal distribution. Show that the distribution of the sample mean bar(X)=(1)/(n)(X_(1)+X_(2)+cdots+X_(n)) has mean mu and variance sigma_( bar(X))^(2)=sigma^(2)//n. [4 marks] (b) A company wishes to establish that the mean lifetime of a certain product is more than 100 days. A sample of size 35 was selected and the products tested. (i) Formulate the null and alternative hypotheses. [4 marks] (ii) The sample mean and sample standard deviation were found to be bar(x)=104.1 days and S=18 days, respectively. What is the conclusion of the hypothesis test at alpha=0.05 ? [6 marks] (iii)