Answer to Question #317908 in Statistics and Probability for Agl

Question #317908

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.

1
Expert's answer
2022-03-28T04:58:50-0400

"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}}\\\\"


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