Answer to Question #343483 in Statistics and Probability for jet

Question

A population consists of values (2, 3, 5). Consider all possible samples of size n=2 that can be drawn with

replacement from this population.

a. Find the mean of the population

b. Find the standard deviation of the population.

c. Find the mean of the sampling distribution of means.

d. Find the standard deviation of the sample distribution of means.

e. Construct the probability histogram of 𝑥̅ with replacement.

Answer 1

a. Population mean

$μ = \frac{2+3+5}{3} = 3.33$

b. The population variance is the sum of squared deviations from the mean divided by n:

$σ^2 = \frac{\sum (x-μ)^2}{n} \\ = \frac{(2 -3.33)^2 + (3 -3.33)^2 +(5-3.33)^2}{3} \\ = \frac{1.7689 + 0.1089 + 2.7889}{3} \\ = \frac{4.6667}{3} \\ = 1.5555$

The population standard deviation is the square root of the population variance:

$σ = \sqrt{σ^2} = \sqrt{1.5555}=1.247$

c.

The mean of the sampling distribution of the mean is the mean of the population from which the scores were sampled. Therefore, the mean of the sampling distribution of means = $\mu$ =3.33. We can also check it:$\frac{2+2.5+3.5+2.5+3+4+3.5+4+5}9=3.33$

d.

$n=9 \\ σ^2 = \frac{\sum (x-μ)^2}{n} \\ = \frac{7}{9} \\ = 0.777 \\ σ = \sqrt{0.777} = 0.882$

The standard deviation of the sample distribution of means = 0.882

e.