1. Explain the importance of the central limit theorem and how this relates to the sample mean.

2. Find the probability of the distribution using the mean x of a random sample of size 65 drawn from a population with a mean 112 and standard deviation 40. Show your work and find the following:

• Find the mean and standard deviation of X
• Find the probability that X assumes a value between 110 and 114.
• Find the probability that X assumes a value greater than 113.

3. Find the probability when we know the mean and standard deviation for a random sample, as per the central limit theorem. The numerical population of grade point averages at a college has mean 2.73 and standard deviation 0.5.

If a random sample of size 110 is taken from the population, what is the probability that the sample mean will be between 2.51 and 2.71? Show the work and explain your answer.