Suppose we have a dataset {x1, 22, . .. ,220}, which satisfies )2 i-1 Xi = 4.0 and 2 x2 = 12.4. (a) (3pt) Find the sample mean a and the sample variance s (note : $2 = n - (Er-1 27 -nx2)). (b) Let's assume a normal model for the population. In other words, we assume that our data is a realization of X1, X2, . .., X20 ~ N(u, 02) for some u and o2 (both parameters are unknown). (i) (4pt) Find a (nontrivial) approximate 95% confidence interval for the population mean / (ii) (4pt) Test Ho : M = 0 vs H1 : / > 0 at level 0.05.(c) After checking the histogram and the boxplot, it turned out that the distribution is approx- imately uniform, and the range(max - min) is about 2. So we decide to apply the following uniform model instead: X1, X2, . . . , X20 ~ Unif([M - 1, Me + 1]) Recall that the mean and the variance of Unif([a, b)] distribution are given by 20 and 12 (b-a) 2, respectively. (i) (3pt) Find the mean and the variance of Unif( u - 1, u + 1]) distribution. (ii) (4pt) Construct a (nontrivial) approximate 95% confidence interval for M. (iii) (4pt) Test Ho : u = 0 vs H1 : / > 0 at level 0.05. (iv) (2pt) Can you trust the result? Briefly explain