Question

Samples of three cards are drawn at random

from a population of eight cards numbered

from 1 to 8.

1. Construct a histogram of the sampling

distribution of the means.

Answer 1

We have population values 1,2,3,4,5,6,7,8, population size N=8 and sample size n=3.

Mean of population $(\mu)$ = $\dfrac{1+2+3+4+5+6+7+8}{8}=4.5$

b.Variance of population 

$$\sigma^2=\dfrac{\Sigma(x_i-\bar{x})^2}{n}=\dfrac{1}{8}(12.25+6.25+2.25+0.25$$ $$+0.25+2.25+6.25+12.25)=5.25$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{5.25}\approx2.29$$

Select a random sample of size 2 without replacement. We have a sample distribution of sample mean.

The number of possible samples which can be drawn without replacement is $^{N}C_n=^{8}C_3=56.$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} no & Sample & Sample \\ & & mean\ (\bar{x}) \\ \hline 1 & 1,2,3 & 6/3 \\ \hdashline 2 & 1,2,4 & 7/3 \\ \hdashline 3 & 1,2,5 & 8/3 \\ \hdashline 4 & 1,2,6 & 9/3 \\ \hdashline 5 & 1,2,7 & 10/3 \\ \hdashline 6 & 1,2,8 & 11/3 \\ \hdashline 7 & 1,3,4 & 8/3 \\ \hdashline 8 & 1,3,5 & 9/3 \\ \hdashline 9 & 1,3,6& 10/3 \\ \hdashline 10 & 1,3,7 & 11/3 \\ \hdashline 11 & 1,3,8 & 12/3 \\ \hdashline 12 & 1,4,5 & 10/3 \\ \hdashline 13 & 1,4,6 & 11/3 \\ \hdashline 14 & 1,4,7 & 12/3 \\ \hdashline 15 & 1,4,8 & 13/3 \\ \hdashline 16 & 1,5,6 & 12/3 \\ \hdashline 17 & 1,5,7 & 13/3 \\ \hdashline 18 & 1,5,8 & 14/3 \\ \hdashline 19 & 1,6,7 & 14/3 \\ \hdashline 20 & 1,6,8 & 15/3 \\ \hdashline 21 & 1,7,8 & 16/3 \\ \hdashline 22 & 2,3,4 & 9/3 \\ \hdashline 23 & 2,3,5 & 10/3 \\ \hdashline 24 & 2,3,6 & 11/3 \\ \hdashline 25 & 2,3,7 & 12/3 \\ \hdashline 26 & 2,3,8 & 13/3 \\ \hdashline 27 & 2,4,5 & 11/3 \\ \hdashline 28 & 2,4,6 & 12/3 \\ \hdashline 29 & 2,4,7 & 13/3 \\ \hdashline 30 & 2,4,8 & 14/3 \\ \hdashline 31 & 2,5,6 & 13/3 \\ \hdashline 32 & 2,5,7 & 14/3 \\ \hdashline 33 & 2,5,8 & 15/3 \\ \hdashline 34 & 2,6,7 & 15/3 \\ \hdashline 35 & 2,6,8 & 16/3 \\ \hdashline 36 & 2,7,8 & 17/3 \\ \hdashline 37 & 3,4,5 & 12/3 \\ \hdashline 38 & 3,4,6 & 13/3 \\ \hdashline 39 & 3,4,7 & 14/3 \\ \hdashline 40 & 3,4,8 & 15/3 \\ \hdashline 41 & 3,5,6 & 14/3 \\ \hdashline 42 & 3,5,7 & 15/3 \\ \hdashline 43 & 3,5,8 & 16/3 \\ \hdashline 44 & 3,6,7 & 16/3 \\ \hdashline 45 & 3,6,8 & 17/3 \\ \hdashline 46 & 3,7,8 & 18/3 \\ \hdashline 47 & 4,5,6 & 15/3 \\ \hdashline 48 & 4,5,7 & 16/3 \\ \hdashline 49 & 4,5,8 & 17/3 \\ \hdashline 50 & 4,6,7& 17/3 \\ \hdashline 51 & 4,6,8 & 18/3 \\ \hdashline 52 & 4,7,8 & 19/3 \\ \hdashline 53 & 5,6,7 & 18/3 \\ \hdashline 54 & 5,6,8 & 19/3 \\ \hdashline 55 & 5,7,8 & 20/3 \\ \hdashline 56 & 6,7,8 & 21/3 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:} \bar{X} & f(\bar{X}) \\ \hline 6/3 & 1/56 \\ \hdashline 7/3 & 1/56 \\ \hdashline 8/3 & 2/56 \\ \hdashline 9/3 & 3/56 \\ \hdashline 10/3 & 4/56 \\ \hdashline 11/3 & 5/56 \\ \hdashline 12/3 & 6/56 \\ \hdashline 13/3 & 6/56 \\ \hdashline 14/3 & 6/56 \\ \hdashline 15/3 & 6/56 \\ \hdashline 16/3 & 5/56 \\ \hdashline 17/3 & 4/56 \\ \hdashline 18/3 & 3/56 \\ \hdashline 19/3 & 2/56 \\ \hdashline 20/3 & 1/56 \\ \hdashline 21/3 & 1/56 \\ \hdashline \end{array}$$