Answer to Question #343722 in Statistics and Probability for Yen

Question

The population of each room in a classroom consist of the numbers 10, 11, 15, and 9. List all the sample size of 3 from this population and compute the mean of each sample

Answer 1

We have population values 9,10,11,15, population size N=4 and sample size n=3.

Mean of population $(\mu)$ = $\dfrac{9+10+11+15}{4}=11.25$

Select a random sample of size 3 without replacement. We have a sample distribution of sample mean.

The number of possible samples which can be drawn without replacement is $^{N}C_n=^{4}C_3=4.$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} no & Sample & Sample \\ & & mean\ (\bar{x}) \\ \hline 1 & 9,10,11 & 30/3 \\ \hdashline 2 & 9,10,15 & 34/3 \\ \hdashline 3 & 9,11,15 & 35/3 \\ \hdashline 4 & 10,11,15 & 36/3 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} \bar{X} & f(\bar{X}) &\bar{X} f(\bar{X}) & \bar{X}^2f(\bar{X}) \\ \hline 30/3 & 1/4 & 30/12 & 900/36 \\ \hdashline 34/3 & 1/4 & 34/12 & 1156/36 \\ \hdashline 35/3 & 1/4& 35/12 & 1225/36 \\ \hdashline 36/3 & 1/4 & 36/12 & 1296/36 \\ \hdashline \end{array}$$

Mean of sampling distribution 

$$\mu_{\bar{X}}=E(\bar{X})=\sum\bar{X}_if(\bar{X}_i)=\dfrac{135}{12}=11.25=\mu$$