This problem will lead you through the steps of determining the probability distribution, mean, and variance of the random variable ¯x. Consider the population made up of only three individuals with corresponding values 1, 2, and 4. The specific sampling procedure you will use involves randomly selecting a sample of size n = 2 from the population with replacement. Sampling with replacement means that each time an individual is drawn from the population, its value is recorded and then the individual is returned to the pool of individuals available selection on the next draw. A with replacement sampling scheme makes it possible for one individual to be selected into the sample multiple times. When sampling with replacement, it is possible to determine the sampling distribution of the sample mean with sample size equal to or larger than the population size. For instance, you could do this same exercise for n = 3 or n = 4.

Define the random variable ¯x as the mean value of the two individuals randomly selected from the population. Write out the possible values of ¯x and their associated probabilities in the form of probability distribution table and sketch its probability histogram. (There are six possible values of ¯x.) Then calculate the mean, µx¯, and standard deviation, σx¯.Verify that µx¯ = µX and σx¯ = σX/ √ n are true statements in this context.