Answer to Question #329326 in Statistics and Probability for Ken ken

Question

Given a die, it has 6 faces in which each face has either dot/s of 𝑥=1,2,3,4,5 𝑎𝑛𝑑 6. Given it as the population, consider sample of size 𝑛=3. Find the population mean, population variance, population standard deviation, the mean, variance and standard deviation of the sampling distribution of the sample mean and illustrate its probability histogram of the sampling distribution of the sample means

Answer 1

$\mu =\frac{1+2+3+4+5+6}{6}=3.5\\\sigma ^2=\frac{\left( 1-3.5 \right) ^2+\left( 2-3.5 \right) ^2+\left( 3-3.5 \right) ^2+\left( 4-3.5 \right) ^2+\left( 5-3.5 \right) ^2+\left( 6-3.5 \right) ^2}{6}=2.91667\\\sigma =\sqrt{\sigma ^2}=1.70783\\\mu _{\bar{x}}=\mu =3.5\\{\sigma _{\bar{x}}}^2=\frac{\sigma ^2}{n}\frac{N-n}{N-1}=\frac{2.91667}{3}\frac{6-3}{6-1}=0.583333\\\sigma _{\bar{x}}=\sqrt{{\sigma _{\bar{x}}}^2}=\sqrt{0.583333}=0.763762$