Question

A simple random sample of size n=14 is obtained from a population with = 63 and a = 16.

(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean?
Assuming that this condition is true, describe the sampling distribution of x.
(b) Assuming the normal model can be used, determine P(x <66.7).
(c) Assuming the normal model can be used, determine P(x ≥ 65.2).

Answer 1

(a) In order to use the normal model to compute probabilities involving the sample mean, the distribution of the population should be approximately normal or at least symmetric. This assumption 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. We can determine the mean and standard deviation of the sampling distribution using the formulas:

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 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 can find that P(z < 0.83) is approximately 0.7967.

(c) To find P(x ≥ 65.2), we can again 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 can find that P(z ≥ 0.60) is approximately 0.7257.