Answer:
(a) In order to use the normal model to compute probabilities involving the sample mean, the population distribution should be approximately normal or at least symmetric. This allows us to rely on the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
(b) Since the population is approximately normal, the sampling distribution of the sample mean will also be approximately normal. To determine probabilities, we need to calculate the mean and standard deviation of the sampling distribution. In this case:
Mean of the sampling distribution = Mean of the population = μ = 63
Standard deviation of the sampling distribution = Standard deviation of the population / Square root of the sample size = σ / √n = 16 / √14 ≈ 4.28
To find the probability P(x < 66.7), we need to standardize the value using the z-score formula:
z = (x - μ) / (σ / √n) = (66.7 - 63) / (4.28) ≈ 0.83
Using a standard normal distribution table or a calculator, we find that P(z < 0.83) is approximately 0.7967.
(c) To find the probability P(x ≥ 65.2), we standardize the value using the z-score formula:
z = (x - μ) / (σ / √n) = (65.2 - 63) / (4.28) ≈ 0.60
Using the standard normal distribution table or a calculator, we find that P(z ≥ 0.60) is approximately 0.7257.