Question

A random sample of 64 observations is drawn from a population with a mean equal to 71 and a standard deviation equal to 16.

What is the mean and the standard deviation of the sampling distribution of x̄?

μ =

σ =

Describe the shape of the sampling distribution of x̄. Does this answer depend on the sample size?

The shape is that of a distribution and depend on the sample size.

Calculate the z-score corresponding to a value of x̄ = 73.8.

Calculate the z-score corresponding to a value of x̄ = 74.2.

Find P(x̄ ≥ 73.8) (to 4 decimals) Find P(x̄ < 74.2) (to 4 decimals)

Find P(73.8 ≤ x̄ ≤ 74.2) (to 4 decimals)

There is a 60% chance that the value of x̄ is above (to 4 decimals).

Answer 1

Mean = = 71

Standard deviation = = 16

The mean and the standard deviation of the sampling distribution of x̄.

= 71

Standard deviation is

The shape of the sampling distribution of is bell shape and it is going to normal distribution.

Yes, the shape of the distribution depends on the sample size.

The z-score corresponding to a value of = 73.8 is

The z-score corresponding to a value of = 74.2 is

P(x̄ ≥ 73.8) = P(Z ≥ 1.4) = 1 - P(Z < 1.4) = 1 - 0.9192 = 0.0808

P(x̄ < 74.2) = P(Z < 1.6) = 0.9452

P(73.8 ≤ x̄ ≤ 74.2) = P(Z < 1.6) - P(Z < 1.4) = 0.9452 - 0.9192 = 0.0260