This problem will explore the random variable X, the sample mean. Consider a population made up of only three individuals with corresponding values 1, 2, and 4. All samples will be drawn with replacement, meaning 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 for selection on the next draw.

1. Consider the scenario where a single individual is randomly selected from the population. Let the random variable X denote the observed value of the one selected individual. Write out the possible values of X and their associated probabilities in a probability distribution table.

2. Calculate the mean, X, and standard deviation, X.

3. Consider the scenario where a sample of n = 2 individuals is randomly selected from the population with replacement. Write out each of the possible outcomes (pairs of observed values) and their associated probabilities in a probability distribution table. (There are nine possible outcome pairs.)

4. Let the random variable X denote the mean value of the two individuals randomly selected from the population. Write out the possible values of X and their associated probabilities in a probability distribution table. (There are six possible values of X.)

5. Calculate the mean, X , and standard deviation, X .

6. Verify that X = X and X = X/ n are true statements in this context.