Question

Consider the population consisting of the values (4, 6, 8).

a. List down all the possible samples of size 2 with replacement.

b. Compute the mean of each sample.

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

d. Identify the probability of each sample.

e. Compute the population mean.

f. Compare the population mean with the mean of the sampling distribution of means.

Answer 1

$a:\\\left( 4,4 \right) ,\left( 4,6 \right) ,\left( 4,8 \right) ,\left( 6,4 \right) ,\left( 6,6 \right) ,\left( 6,8 \right) ,\left( 8,4 \right) ,\left( 8,6 \right) ,\left( 8,8 \right) \\b:\\\left( 4,4 \right) ,\bar{x}=4\\\left( 4,6 \right) ,\bar{x}=5\\\left( 4,8 \right) ,\bar{x}=6\\\left( 6,4 \right) ,\bar{x}=5\\\left( 6,6 \right) ,\bar{x}=6\\\left( 6,8 \right) ,\bar{x}=7\\\left( 8,4 \right) ,\bar{x}=6\\\left( 8,6 \right) ,\bar{x}=7\\\left( 8,8 \right) ,\bar{x}=8\\c:\\\mu _{\bar{x}}=\frac{4+2\cdot 5+3\cdot 6+2\cdot 7+8}{9}=6\\d:\\P\left( \left( 4,4 \right) \right) =P\left( \left( 4,6 \right) \right) =P\left( \left( 4,8 \right) \right) =P\left( \left( 6,4 \right) \right) =P\left( \left( 6,6 \right) \right) =P\left( \left( 6,8 \right) \right) =P\left( \left( 8,4 \right) \right) =P\left( \left( 8,6 \right) \right) =P\left( \left( 8,8 \right) \right) =\frac{1}{9}\\e:\\\mu =\frac{4+6+8}{3}=6\\f:\\\mu =\mu _{\bar{x}}$