Problem 3 (R - 19 points). You have learned in probability that the sampling distribution of Xis not normally distribute

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Problem 3 R 19 Points You Have Learned In Probability That The Sampling Distribution Of Xis Not Normally Distribute 1 (134.57 KiB) Viewed 41 times

Problem 3 R 19 Points You Have Learned In Probability That The Sampling Distribution Of Xis Not Normally Distribute 2 (96.63 KiB) Viewed 41 times

Problem 3 (R - 19 points). You have learned in probability that the sampling distribution of Xis not normally distributed when X; ~ N(H1,04). However, Central Limit Theorem tells us that the distribution should converge to a normal distribution in the limit as n, the sample size, grows to infinity. In this problem, you will investigate this visually, and by comparing probabilities calculated from CLT and from simulation. Parts (a), (b) require plotting, but all other parts can be completed independently without these graphs. The Weibull distribution is defined by two parameters: shape (a > 0) and scale (>0) and has the following density function: CE f(x]a,b) = В 0-1 e-(1/8) >0 Weibull distributed random variables have the following expected value and variance: E(X) = B. 1(1+1/a) V(X) = 32. ((1+2/a) – (T(1+1/a))?) [(y) is the Gamma function, and can be tricky to compute if y is not a positive integer. Instead, you can calculate the values using R using the gamma(y) function. See ?gamma for usage information. 4 a) (2 points) Plot a graph in R of the density function of Weibull(a = 0.5, B = 2), compute the mean of this distribution (provided above) in R and plot it as a vertical line in your graph. Briefly describe the shape, skew (if any) of the distribution, and explain whether you think that the result of CLT will apply with a sample size of 15. b) (6 points) Using a simulation size of 1000 and your student number as the seed, for each sample size starting at n = 20 and increasing to n = 120 in increments of 20, produce and save the corresponding density histogram representing the sampling distribution of Xn. Based on visual inspection, state which sample size you think is sufficiently large for Central Limit Theorem to offer good approximations. Your plots should include in their titles: the parameters of the Weibull distribution, the sample size, and the simulation size. Display all plots in a grid using grid.arrange.

Problem 3 (R - 19 points). You have learned in probability that the sampling distribution of Xn is not normally distributed when X; * N (1,0). However, Central Limit Theorem tells us that the distribution should converge to a normal distribution in the limit as n, the sample size, grows to infinity. In this problem, you will investigate this visually, and by comparing probabilities calculated from CLT and from simulation. Parts (a)- (b) require plotting, but all other parts can be completed independently without these graphs. The Weibull distribution is defined by two parameters: shape (a > 0) and scale (B > 0) and has the following density function: Q-1 а f(xla, 3) = ; (?). | e-(x/B)", > 0 Weibull distributed random variables have the following expected value and variance: E(X) = B. 1(1+1/a) V(X) = B2. (I (1 + 2/a) – (1(1+1/a))?) - [(y) is the Gamma function, and can be tricky to compute if y is not a positive integer. Instead, you can calculate the values using R using the gamma(y) function. See ?gamma for usage information.