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Variance estimation is an usual interest of statisticans. It helps to construct confidence in- tervals for estimations. An estimator for variance o' could be the sample variance S,. mi EL,(Y: - Y)? where Y = 1 ELY. It is known that the random variable ("-1)s has the x2 1 distribution with degree of freedom n - 1 if Y1, ..., Y, are independent and identi- cally distributed with the Normal(/, ?) distribution. That is also to say the random variable ("-1)s- has the Gamma(,], ;) distribution. We will start from Normal approximation to Gamma using the CLT. Let X1, ..., Xn are independent and identically distributed with the Gamma(a, 8) distribution. Then we have X = 1ELIXQ, #x = E[X] = g and of = Var(X) = 14 no. We could approximate the probability P(X y) = with a = 6, 8 = 3 and n = 40 (Give your answer up to 4 decimal places). QUESTION 10 : Take n = 40 and o? = 1. Using the CLT, approximate the quantile c, that the probability P(S, c2) = 0.025. (Give your answer up to 4 decimal places)