Question 6 - Central Limit Theorem (15 Marks) Sampling Process: Assume that we randomly select samples of the same size , an Infinite number of times from a population that follows a Poleson distribution with mean off ), and then, we calculate the mean of scores In each sample. Question 6.a (2 Marks) What does Central Limit Theorem tell us about the sampling distribution of the sample mean? YOUR ANSWER HERE Question 6.b (3 Marks) For three different Poleson populations with mean of 21 - 1. A; - 5 and /} - 20, we will do the sampling four separate times -- for small samples (n-10), for samples of 100 subjects (n-100) and 1000 subjects (n-1000). and once for big samples (n-10000) Based on your answer from 6.3, compute the parameter values for each sampling distribution In R. YOUR ANSWER HERE Question 6.c (5 Marks) In this question, you sire asked to experimentally justify the result In the CLT Theorem. For different sample sizes of , - 10, 100 and 1000, use 50000 simulations (Le. to approximate the Infinite times we drew samples as mentioned before) to Implement the sampling process. From those 50000 sample means, compute the mean and standard deviation parameters (3 sample sizes and 3 / rates, 9 pairs of parameters In total). Discuss how the results reflect the CLT. Plot the results ( mean and standard deviation separately) to demonstrate any effects you want to discuss. YOUR ANSWER HERE Question 6.d (5 Marks) When rate 1, - 1 and My - 5 and sample size w l6 10 or 100, obtain the z scores of the sample means (from 5000 0 simulations). Plot their distributions In a histogram with the theoretical Gaussian curve overiald. Note that for sample size 100, the plots overlay very nicely. But what happens with sample size 10? Explain the differences between the four plots. For each simulation: the z score of the mean can be calculated as (X - #) where X Is the mean of the sample. or Is the population mean and of Is the population standard deviation. YOUR ANSWER HERE