Answer to Question #322862 in Statistics and Probability for Nate

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 (μ) and standard deviation (σ) of the population?

2. How many different samples of size n = 4 can be drawn from the population? List them with their corresponding means.

Sample Mean

1.

2.

3.

4.

5.

3. Construct the sampling distribution of the sample means.

Sample Mean𝐗̅ Frequency Probability(𝐗̅)

1.

2.

3.

4.

5.

6.

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

5. What is the standard deviation (σX̅) of the sampling distribution of the sample means? Compare this to the standard deviation of the population.

Answer 1

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