Question

Random samples of size 𝑛 = 2 are drawn from a finite population

consisting of the numbers 5,6,7,8, and 9.

a. Find the mean of the population 𝜇.

b. Find the standard deviation of the population 𝜎.

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

d. Find the standard deviation of the sampling distribution of the sample

means 𝜎𝑋̅.

e. Verify the Central Limit theorem by:

I. Comparing 𝜇 and 𝜇𝑋̅.

II. Comparing 𝜎 and 𝜎𝑋̅.

Answer 1

$a:\mu =\frac{5+6+7+8+9}{5}=7\\b:\sigma =\sqrt{\frac{\left( 5-7 \right) ^2+\left( 6-7 \right) ^2+\left( 7-7 \right) ^2+\left( 8-7 \right) ^2+\left( 9-7 \right) ^2}{5}}=\sqrt{2}\\c:All\,\,samples\\\left( 5,6 \right) ,\bar{x}=5.5\\\left( 5,7 \right) ,\bar{x}=6\\\left( 5,8 \right) ,\bar{x}=6.5\\\left( 5,9 \right) ,\bar{x}=7\\\left( 6,7 \right) ,\bar{x}=6.5\\\left( 6,8 \right) ,\bar{x}=7\\\left( 6,9 \right) ,\bar{x}=7.5\\\left( 7,8 \right) ,\bar{x}=7.5\\\left( 7,9 \right) ,\bar{x}=8\\\left( 8,9 \right) ,\bar{x}=8.5\\\mu _{\bar{x}}=\frac{5.5+6+2\cdot 6.5+2\cdot 7+2\cdot 7.5+8+8.5}{10}=7\\d: \sigma _{\bar{x}}=\sqrt{\frac{\left( 5.5-7 \right) ^2+\left( 6-7 \right) ^2+2\cdot \left( 6.5-7 \right) ^2+2\left( 7-7 \right) ^2+2\left( 7.5-7 \right) ^2+\left( 8-7 \right) ^2+\left( 8.5-7 \right) ^2}{10}}=\sqrt{0.75}\\e:\\I:\mu =\mu _{\bar{x}}\\II:\frac{\sigma}{\sqrt{2}}\sqrt{\frac{5-2}{5-1}}=\frac{\sqrt{2}}{\sqrt{2}}\sqrt{\frac{3}{4}}=\sqrt{0.75}=\sigma _{\bar{x}}\\$