Answer to Question #323229 in Statistics and Probability for Patrick

Question

Given the population 3, 5, 8, 9, and 10. Suppose samples of size 4 are drawn from this population

1. What is the mean (u) and standard deviation (a) of the population?

2. How many different samples of size n& can be drawn from the population? List them

corresponding means. 3. Construct the sampling distribution of the sample means

4 What is the mean () of the sampling distribution of the sample means? Compare this to the mean

the population

5. What is the standard deviation (n) of the sampling distribution of the sample means? Compare thin

the standard deviation of the population.

Answer 1

$1:\mu =\frac{3+5+8+9+10}{5}=7\\\sigma =\sqrt{\frac{\left( 3-7 \right) ^2+\left( 5-7 \right) ^2+\left( 8-7 \right) ^2+\left( 9-7 \right) ^2+\left( 10-7 \right) ^2}{5}}=\sqrt{6.8}\\2:\\C_{5}^{4}=5\\\left( 3,5,8,9 \right) ,\bar{x}=\frac{3+5+8+9}{4}=6.25\\\left( 3,5,8,10 \right) ,\bar{x}=\frac{3+5+8+10}{4}=6.5\\\left( 3,5,9,10 \right) ,\bar{x}=\frac{3+5+9+10}{4}=6.75\\\left( 3,8,9,10 \right) ,\bar{x}=\frac{3+8+9+10}{4}=7.5\\\left( 5,8,9,10 \right) ,\bar{x}=\frac{5+8+9+10}{4}=8\\3:\\P\left( \bar{x}=6.25 \right) =P\left( \bar{x}=6.5 \right) =P\left( \bar{x}=6.75 \right) =P\left( \bar{x}=7.5 \right) =P\left( \bar{x}=8 \right) =0.2\\4:\\\mu _{\bar{x}}=\frac{6.25+6.5+6.75+7.5+8}{5}=7=\mu \\5:\\\sigma _{\bar{x}}=\sqrt{\frac{\left( 6.25-7 \right) ^2+\left( 6.5-7 \right) ^2+\left( 6.75-7 \right) ^2+\left( 7.5-7 \right) ^2+\left( 8-7 \right) ^2}{5}}=\\=\sqrt{0.425}=\frac{\sqrt{6.8}}{\sqrt{4}}\sqrt{\frac{5-4}{5-1}}=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N-1}}$