Answer to Question #199431 in Statistics and Probability for ahmad

Question

Given the population 5, 10, 15, 20, 25 a) How many samples of size 3, can be drawn with replacement from this population b) Compute and tabulate the sampling distribution of the mean from samples of size 3. c) Verify the results of mean and variance of sampling distribution of the mean. 

Answer 1

Mean

$$\mu=\dfrac{5+10+15+20+25}{5}=15$$

Variance

$$\sigma^2=\dfrac{1}{5}\big((5-15)^2+(10-15)^2+(15-15)^2$$ $$(20-15)^2+(25-15)^2\big)=50$$

Standard deviation

$$\sigma=\sqrt{\sigma^2}=\sqrt{50}=5\sqrt{2}\approx7.071068$$

We have population values $5,10,15,20,25$ population size $N=5$ and sample size $n=3.$ Thus, the number of possible samples which can be drawn with replacement is

$$n^r=5^3=125$$ $$\def\arraystretch{1.5} \begin{array}{c:c:c} Sample & Sample & Sample \ mean \\ No. & values & (\bar{X}) \\ \hline 1 & 5,5, 5 & 5 \\ \hdashline 2 & 5,5,10 & 20/3 \\ \hdashline 3 & 5,5,15 & 25/3 \\ \hdashline 4 & 5,5,20 & 10 \\ \hdashline 5 & 5,5,25 & 35/3 \\ \hline 6 & 5,10,5 & 20/3 \\ \hline 7 & 5,10,10 & 25/3 \\ \hline 8 & 5,10,15 & 10 \\ \hline 9 & 5,10,20 & 35/3 \\ \hline 10 & 5,10,25 & 40/3 \\ \hline 11 & 5,15,5 & 25/3 \\ \hline 12 & 5,15,10 & 10 \\ \hline 13 & 5,15,15 & 35/3 \\ \hline 14 & 5,15,20 & 40/3 \\ \hline 15 & 5,15,25 & 15 \\ \hline 16 & 5,20,5 & 10 \\ \hline 17 & 5,20,10 & 35/3 \\ \hline 18 & 5,20,15 & 40/3 \\ \hline 19 & 5,20,20 & 15 \\ \hline 20 & 5,20,25 & 50/3 \\ \hline 21 & 5,25,5 & 35/3\\ \hline 22 & 5,25,10 & 40/3 \\ \hline 23 & 5,25,15 & 15 \\ \hline 24 & 5,25,20 & 50/3 \\ \hline 25 & 5,25,25 & 55/3 \\ \hline 26 & 10,5, 5 & 20/3 \\ \hdashline 27 & 10,5,10 & 25/3 \\ \hdashline 28 & 10,5,15 & 10 \\ \hdashline 29 & 10,5,20 & 35/3 \\ \hdashline 30 & 10,5,25 & 40/3 \\ \hline 31 & 10,10,5 & 25/3 \\ \hline 32 & 10,10,10 & 10 \\ \hline 33 & 10,10,15 & 35/3 \\ \hline 34 & 10,10,20 & 40/3 \\ \hline 35 & 10,10,25 & 15 \\ \hline 36 & 10,15,5 & 10 \\ \hline 37 & 10,15,10 & 35/3\\ \hline 38 & 10,15,15 & 40/3 \\ \hline 39 & 10,15,20 & 15 \\ \hline 40 & 10,15,25 & 50/3 \\ \hline 41 & 10,20,5 & 35/3 \\ \hline 42 & 10,20,10 & 40/3 \\ \hline 43 & 10,20,15 & 15 \\ \hline 44 & 10,20,20 & 50/3 \\ \hline 45 & 10,20,25 & 55/3 \\ \hline 46 & 10,25,5 & 40/3\\ \hline 47 & 10,25,10 & 15 \\ \hline 48 & 10,25,15 & 50/3 \\ \hline 49 & 10,25,20 & 55/3 \\ \hline 50 & 10,25,25 & 20 \\ \hline 51 & 15,5, 5 & 25/3 \\ \hdashline 52 & 15,5,10 & 10 \\ \hdashline 53 & 15,5,15 & 35/3 \\ \hdashline 54 & 15,5,20 & 40/3 \\ \hdashline 55 & 15,5,25 & 15 \\ \hline 56 & 15,10,5 & 10 \\ \hline 57 & 15,10,10 &35/3 \\ \hline 58 & 15,10,15 & 40/3 \\ \hline 59 & 15,10,20 & 15 \\ \hline 60 & 15,10,25 & 50/3 \\ \hline 61 & 15,15,5 & 35/3 \\ \hline 62 & 15,15,10 & 40/3 \\ \hline 63 & 15,15,15 & 15 \\ \hline 64 & 15,15,20 & 50/3 \\ \hline 65 & 15,15,25 & 55/3 \\ \hline 66 & 15,20,5 & 40/3 \\ \hline 67 & 15,20,10 & 15 \\ \hline 68 & 15,20,15 & 50/3 \\ \hline 69 & 15,20,20 & 55/3 \\ \hline 70 & 15,20,25 & 20 \\ \hline 71 & 15,25,5 & 15\\ \hline 72 & 15,25,10 & 50/3 \\ \hline 73 & 15,25,15 & 55/3 \\ \hline 74 & 15,25,20 & 20 \\ \hline 75 & 15,25,25 & 65/3 \\ \hline 76 & 20,5, 5 & 10 \\ \hdashline 77 & 20,5,10 & 35/3 \\ \hdashline 78 & 20,5,15 & 40/3 \\ \hdashline 79 & 20,5,20 & 15 \\ \hdashline 80 & 20,5,25 & 50/3 \\ \hline 81 & 20,10,5 & 35/3 \\ \hline 82 & 20,10,10 & 40/3 \\ \hline 83 & 20,10,15 & 15 \\ \hline 84 & 20,10,20 & 50/3 \\ \hline 85 & 20,10,25 & 55/3 \\ \hline 86 & 20,15,5 & 40/3 \\ \hline 87 & 20,15,10 & 15\\ \hline 88 & 20,15,15 & 50/3 \\ \hline 89 & 20,15,20 & 55/3 \\ \hline 90 & 20,15,25 & 20 \\ \hline 91 & 20,20,5 & 15 \\ \hline 92 & 20,20,10 & 50/3 \\ \hline 93 & 20,20,15 & 55/3 \\ \hline 94 & 20,20,20 & 20 \\ \hline 95 & 20,20,25 & 65/3 \\ \hline 96 & 20,25,5 & 50/3\\ \hline 97 & 20,25,10 & 55/3 \\ \hline 98 & 20,25,15 & 20 \\ \hline 99 & 20,25,20 & 65/3 \\ \hline 100 & 20,25,25 & 70/3 \\ \hline 101 & 25,5, 5 & 35/3 \\ \hdashline 102 & 25,5,10 & 40/3 \\ \hdashline 103 & 25,5,15 & 15 \\ \hdashline 104 & 25,5,20 & 50/3 \\ \hdashline 105 & 25,5,25 & 55/3 \\ \hline 106 & 25,10,5 & 40/3 \\ \hline 107 & 25,10,10 & 15 \\ \hline 108 & 25,10,15 & 50/3 \\ \hline 109 & 25,10,20 & 55/3 \\ \hline 110 & 25,10,25 & 20 \\ \hline 111 & 25,15,5 & 15 \\ \hline 112 & 25,15,10 & 50/3 \\ \hline 113 & 25,15,15 & 55/3 \\ \hline 114 & 25,15,20 & 20 \\ \hline 115 & 25,15,25 & 65/3 \\ \hline 116 & 25,20,5 & 50/3 \\ \hline 117 & 25,20,10 & 55/3 \\ \hline 118 & 25,20,15 & 20 \\ \hline 119 & 25,20,20 & 65/3 \\ \hline 120 & 25,20,25 & 70/3 \\ \hline 121 & 25,25,5 & 55/3\\ \hline 122 & 25,25,10 & 20 \\ \hline 123 & 25,25,15 & 65/3 \\ \hline 124 & 25,25,20 & 70/3 \\ \hline 125 & 25,25,25 & 25 \\ \hline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} \bar{X} & f & f(\bar{X}) & \bar{X}f(\bar{X})& \bar{X}^2f(\bar{X}) \\ \hline 15/3 & 1& 1/125 & 3/75 & 9/45 \\ \hdashline 20/3 & 3 & 3/125 & 12/75 & 48/45 \\ \hdashline 25/3 & 6 & 6/125 & 30/75 & 150/45 \\ \hdashline 30/3 & 10 & 10/125 & 60/75 & 360/45 \\ \hdashline 35/3 & 15 & 15/125 & 105/75& 735/45 \\ \hdashline 40/3 & 18 & 18/125 & 144/75 & 1152/45 \\ \hdashline 45/3 & 19 & 19/125 & 171/75 & 1539/45 \\ \hdashline 50/3 & 18 & 18/125 & 180/75 & 1800/45 \\ \hdashline 55/3 & 15 & 15/125 & 165/75 & 1815/45 \\ \hdashline 60/3 & 10 & 10/125 & 120/75 & 1440/45 \\ \hdashline 65/3 & 6 & 6/125 & 78/75 & 1014/45 \\ \hdashline 70/3 & 3 & 3/125 & 42/75 & 588/45 \\ \hdashline 75/3 & 1& 1/125 & 15/75 & 225/45 \\ \hdashline Total & 125 & 1 & 225/15 & 725/3 \\ \hline \end{array}$$

$$E(\bar{X})=\sum\bar{X}f(\bar{X})=\dfrac{225}{15}=15$$

The mean of the sampling distribution of the sample means is equal to the

the mean of the population.

$$E(\bar{X})=15=\mu$$

$$Var(\bar{X})=\sum\bar{X}^2f(\bar{X})-(\sum\bar{X}f(\bar{X}))^2$$

$$=\dfrac{725}{3}-(15)^2=\dfrac{50}{3}$$

$$\sqrt{Var(\bar{X})}=\sqrt{\dfrac{50}{3}}\approx4.082483$$

Verification:

$$Var(\bar{X})=\dfrac{\sigma^2}{n}=\dfrac{50}{3},True$$