Answer to Question #333904 in Statistics and Probability for shiira

Question

Consider a population consisting of 3, 6, 7, 9, 10, 4, and 8. Suppose samples of size 2 are drawn from this population. Describe the sampling distribution of sample means. What is the standard deviation of the sampling distribution? NOTE: EXPRESS YOUR ANSWER INTO 4 DECIMAL PLACE VALUES

Answer 1

Here population size : $N =7$ and we have to draw a sample of size 2

So, there are $^{N}C_n$ possible samples that is, $^{7}C_2=21$

Mean of population $(\mu)$ = $\dfrac{3+6+7+9+10+4+8}{7}=\dfrac{47}{7}\approx6.7143$

Variance of population 

$$(\sigma^2)=\dfrac{\Sigma(x_i-\bar{x})^2}{n}=\dfrac{1}{7}(\dfrac{676}{49}+\dfrac{25}{49}+\dfrac{4}{49}$$

$$+\dfrac{256}{49}+\dfrac{529}{49}+\dfrac{361}{49}+\dfrac{81}{49})$$

$$=\dfrac{1932}{343}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} no & Sample & Sample \\ & & mean\ (\bar{x}) \\ \hline 1 & 3,4 & 3.5 \\ \hdashline 2 & 3,6 & 4.5 \\ \hdashline 3 & 3,7 & 5 \\ \hdashline 4 & 3,8 & 5.5 \\ \hdashline 5 & 3,9 & 6 \\ \hdashline 6 & 3,10 & 6.5 \\ \hdashline 7 & 4,6 & 5 \\ \hdashline 8 & 4,7 & 5.5 \\ \hdashline 9 & 4,8 & 6 \\ \hdashline 10 & 4,9 & 6.5 \\ \hdashline 11 & 4,10 & 7 \\ \hdashline 12 & 6,7 & 6.5 \\ \hdashline 13 & 6,8 & 7 \\ \hdashline 14 & 6,9 & 7.5 \\ \hdashline 15 & 6,10 & 8 \\ \hdashline 16 & 7,8 & 7.5 \\ \hdashline 17 & 7,9 & 8 \\ \hdashline 18 & 7,10 & 8.5 \\ \hdashline 19 & 8,9 & 8.5 \\ \hdashline 20 & 8,10 & 9 \\ \hdashline 21 & 9,10 & 9.5 \\ \hdashline \end{array}$$

Mean of sampling distribution 

$$\mu_{\bar{x}}=E(\bar{x})=\sum\bar{x}_if(\bar{x}_i)=\dfrac{47}{7}=\mu$$

The variance of sampling distribution 

$$Var(\bar{x})=\sigma^2_{\bar{x}}=\sum\bar{x}_i^2f(\bar{x}_i)-\big[\sum\bar{x}_if(\bar{x}_i)\big]^2$$ $$=\dfrac{332}{7}-\dfrac{2209}{49}=\dfrac{115}{49}= \dfrac{\sigma^2}{n}(\dfrac{N-n}{N-1})$$

$$\sigma_{\bar{x}}=\sqrt{\dfrac{115}{49}}\approx1.5320$$