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Explain the central limit theorem in the context of the

1. It is known that the population of students who take EconS 311 have a final grade that is normally distributed with µ_x = 81 and s_x² = 25.

a. For one observation (student) from this population, find Pr (X < 80).

Now suppose that a random sample of students is selected from this population. Note that the next two questions are about X‾, not about X itself.

b. In a random sample of size n = 64, find Pr (X < 80).

c. In a random sample of size n = 36, find Pr (78< X < 84).

d. Explain the central limit theorem in the context of the current problem, and indicate whether it was needed to find these probabilities in a, b, and c (might differ for each of a, b, and c).

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