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ANSWER USING R. THANK YOU​​​​​​
2. Consider a random variable X. Its skewness (i.e. 3rd cumulant) is E[(X – ux)] o K3 = If X has a normal distribution, then K3 = 0). Let us assume that we have a random sample X1, ... , Xn from this normal population. The sample skewness is 1 21–1(X; - X) (121–1(X; - X)2)3/2 k3 = It can be shown that K3 Z= N(0,1), where V[R3] 6 (n − 2) (n + 1)(n+3) V V(Â3] The skewness test for normality is to test Ho : K3 = 0 against HQ : K3 = 0, using the above z-test statistic, and rejecting H, in favour of Ha, if |z| > 1.96. Note that since a normal random variable has a skewness of 0, then rejecting H, is interpreted as evidence against normality. Compare the power of the skewness test with that of the Shapiro-Wilk test (i.e shapiro.test with R). Use n = 10, 20, 50, 70, 100. Consider the following distributions. (a) Consider a T(v) distribution for v = 3, 6, 12. (You can use rt.) (b) Consider a gamma(shape, scale 1) distribution for shape = 0.25, 1, 2, 4. (You can use rgamma.)