(a) The sampling distribution of x-bar, the sample mean, from a population follows a normal distribution, regardless of the shape of the population distribution, if the sample size is large enough. This is known as the Central Limit Theorem.
(b) To find P(x-bar > 85), we can convert this into a z-score using the formula: z = (x-bar - μ) / (σ / √n), then find the probability using a standard normal distribution table.
(c) To find P(x-bar ≤ 76.2), we can also convert this into a z-score using the formula and find the probability using the standard normal distribution table.
(d) To find P(78.7 < x-bar < 84.3), we calculate the z-scores for both values and find the corresponding probabilities.
