Question

A. Consider a population consisting of values (1,3,5).

1. List all the possible samples of size 2 that can be drawn from the population

with replacement.

Observation Sample X (X − μX) (X − μX)²

2. Compute for the mean of the sampling distribution of the sample means.

3. Compute for the variance of the sampling distribution of the sample means.

X f Probability

P(X)

4. Construct the probability histogram of means with replacements when n = 2.

Answer 1

$1:\\\mu _X=\frac{1+3+5}{3}=3\\2:\\\mu _{\bar{x}}=\frac{1+2+3+2+3+4+3+4+5}{9}=3\\3:\\{\sigma _{\bar{x}}}^2=\frac{\left( 1-3 \right) ^2+2\cdot \left( 2-3 \right) ^2+3\cdot \left( 3-3 \right) ^2+2\cdot \left( 4-3 \right) ^2+\left( 5-3 \right) ^2}{9}=\frac{4}{3}$

We calculate the probabilities as number of correspondent value in the column of means divided by 9.