Answer to Question #336235 in Statistics and Probability for John

Question

A. The following are the heights of four students in centimeters. Suppose

samples of size 2 are taken from this population of four students.

STUDENTS | HEIGHT(in cm)

Cardo 125

Alyana 120

Joaquin 130

Flora 110

a. Compute the mean of the population

b. Compute the mean of the sampling distribution of the sample means

Answer 1

We have population values 125, 120, 130, 110, population size N=4 and sample size n=2.

Mean of population $(\mu)$ = 

$$\dfrac{125+120+130+110}{4}=121.25$$

The number of possible samples which can be drawn without replacement is $^{N}C_n=^{4}C_2=6.$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} no & Sample & Sample \\ & & mean\ (\bar{x}) \\ \hline 1 & 125,120 & 245/2 \\ \hdashline 2 & 125, 130 & 255/2 \\ \hdashline 3 & 125,110 & 235/2 \\ \hdashline 4 & 120,130 & 250/2 \\ \hdashline 5 & 120,110 & 230/2 \\ \hdashline 6 & 130,110 & 240/2 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} \bar{X} & f(\bar{X}) &\bar{X} f(\bar{X}) \\ \hline 230/2 & 1/6 & 230/12\\ \hdashline 235/2 & 1/6 & 235/12\\ \hdashline 240/2 & 1/6 & 240/12\\ \hdashline 245/2 & 1/6 & 245/12 \\ \hdashline 250/2 & 1/6 & 250/12\\ \hdashline 255/2 & 1/6 & 255/12 \\ \hdashline \end{array}$$

Mean of sampling distribution 

$$\mu_{\bar{X}}=E(\bar{X})=\sum\bar{X}_if(\bar{X}_i)=121.25=\mu$$