Question

1. A population of 4 numbers 1,2,4,5

a. List ALL the possible samples of size n=3, which can be drawn with replacement from the population.

b. Construct a sampling distribution of the sample mean.

1. A population consists of 4 numbers 3,7,11,15 of sample size n=2 which can be drawn without replacement from the population.

a. Population mean

b. Population Variance

c. Population standard deviation

d. Mean of the sampling distribution of the sample means

e. Variance of the sampling distribution of the sample means

f. Standard deviation of the sampling distribution of the sample means

Answer 1

$a:\mu =\frac{3+7+11+15}{4}=9\\b:\sigma ^2=\frac{\left( 3-9 \right) ^2+\left( 7-9 \right) ^2+\left( 11-9 \right) ^2+\left( 15-9 \right) ^2}{4}=20\\c:\sigma =\sqrt{\sigma ^2}=\sqrt{20}=4.47214\\d:\mu _{\bar{x}}=\frac{5+7+2\cdot 9+11+13}{6}=9\\e:{\sigma ^2}_{\bar{x}}=\frac{\left( 5-9 \right) ^2+\left( 7-9 \right) ^2+2\cdot \left( 9-9 \right) ^2+\left( 11-9 \right) ^2+\left( 13-9 \right) ^2}{6}=6.66667\\f:\sigma _{\bar{x}}=\sqrt{{\sigma ^2}_{\bar{x}}}=\sqrt{6.666667}=2.58199$