Question

Consider all possible samples of size 4 (𝑛=4) taken with replacement from a population consisting of

the values 1, 3, and 5.

I. Compute the following:

1. Population Mean (μ)

2. Population Variance (𝜎²)

3. Population Standard Deviation (σ)

4. Mean of Sample Mean (μx̅)

5. Variance of Sample Mean (σ²x̅)

6. Standard Deviation of Sample Mean (σx̅)

Answer 1

1.

$\mu=\frac{1+3+5}{3}=3$

2.

$\sigma^2=\frac{\sum(x_i-\mu)^2}{n}=\frac{2^2+2^2}{3}=8/3$

3.

$\sigma=\sqrt{8/3}=1.63$

4.

$\mu(\overline {x})=\frac{1+1.5+2+2.5+3+2+3+4+5+3.5+4+4.5}{12}=2.8$

5.

$\sigma^2(\overline {x})=\frac{1.8^2+1.3^2+0.8^2+0.3^2+0.2^2+0.8^2+0.2^2+0.2^2+2.2^2+0.7^2+1.2^2+1.7^2}{12}=1.34$

6.

$\sigma(\overline {x})=\sqrt{1.34}=1.16$