Question 1 (20%} Consider a population 1with mean In and size N = 5. Generate random samples with replacement. i. How many samples are there for all possible samples of size n = 3? ii. What is the mean of all sample means for all possible samples of size n = 3? Question 2 [40%) Consider the following sample X = {3, 3,1", 1, 5, 4,9,,2,3} from a population of large size. i. What are the sample mean and sample variance of sample X? ii. If someone generates a large number (say, hundreds} of random samples of the same size as X from the same population, can we apply,-r the central limit theorem and mnclude that the sample means will approximately follow a normal distribution? Why? Now assume that population has mean p: = E and variance or2 = 16. iii. What is the standard error of the mean for samples of the same size n = 144? iv. Suppose a random sample of size n = 255 is selected. What is the interval that includes the sample mean [whose distribution is approximately normal]I with probability 95%? Question 3 (40%) i. In class we have learnt that the formula for sample variance is $2 = 2.(X1-x)? Show that n-1 82 = En (X]-x2 ) n-1 ii. In class we have also learnt that the formula for population variance is o' = Show that 02 - 21-1 21,(31-1,)' 1 2n2 Hint: note that A = ', in which i is just an index variable that can also be denoted by j