Question

2. Gillbert got a new job in a shoe store. The number of pairs of shoes he was able to sell for three days are 2, 4, and 5. Assume that samples of size 2 are randomly selected with replacement from this population of three values.

a. List down the 9 different possible samples.

b. Find the mean of each sample.

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

d. Identify the probability of each sample.

e. Find the population mean.

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

Answer 1

We have population values $2,4,5,$ population size $N=3$  and sample size $n=2.$

Thus, the number of possible samples which can be drawn with replacements is $3^2=9.$

a.

b.

$$\def\arraystretch{1.5} \begin{array}{c:c} Sample\ values & Sample\ mean\ (\bar{x}) \\ \hline 2,2 & 2 \\ \hdashline 2,4 & 3 \\ \hdashline 2,5 & 3.5 \\ \hdashline 4,2 & 3 \\ \hdashline 4,4 & 4 \\ \hdashline 4,5 & 4.5 \\ \hdashline 5,2 & 3.5 \\ \hdashline 5,4 & 4.5 \\ \hdashline 5,5 & 5 \end{array}$$

c.

d.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & \bar{x} & f & f(\bar{x}) & \bar{x}f(\bar{x})& \bar{x}^2 f(\bar{x})\\ \hline & 2 & 1 & 1/9 & 2/9 & 4/9 \\ \hdashline & 3 & 2 & 2/9 & 2/3 & 2 \\ \hdashline & 3.5 & 2 & 2/9 & 7/9 & 49/18 \\ \hdashline & 4 & 1 & 1/9 & 4/9 & 16/9 \\ \hdashline & 4.5 & 2 & 2/9 & 1 & 1/2 \\ \hdashline & 5 & 1 & 1/9 & 5/9 & 25/9 \\ \hdashline Sum= & & 9 & 1 & 11/3 & 92/9 \\ \hdashline \end{array}$$

$$\mu_{\bar{x}}=E(\bar{X})=11/3$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 2 & 3 & 3.5 & 4 & 4.5 & 5 \\ \hline p(x) & 1/9 & 2/9 & 2/9 & 1/9 & 2/9 & 1/9 \\ \end{array}$$

e.

$$\mu=\dfrac{2+4+5}{3}=\dfrac{11}{3}$$

f.

$$\mu_{\bar{x}}=11/3=\mu$$