Question

assume a population of 1, 2, and 12. assume that samples of size ( n = 2 ) are randomly selected with replacement from the population. listed below are the nine different samples. complete parts a through d below.
1,1 1,2 1,12 2,1 2,2 2,12 12,1 12,2 12,12
a. find the value of the population standard deviation ( sigma ).
( sigma = 4.067 ) (round to three decimal places as needed.)
b. find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution of the distinct standard deviation values. use ascending order of the sample standard deviations.
s probability
(type integers or fractions.)

Explanation:

Step1: Calculate standard deviation for each sample

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\), where \(n = 2\)

• For sample \((1,1)\): \(\bar{x}=\frac{1 + 1}{2}=1\), \(s=\sqrt{\frac{(1 - 1)^2+(1 - 1)^2}{2-1}} = 0\)
• For sample \((1,2)\): \(\bar{x}=\frac{1+2}{2}=1.5\), \(s=\sqrt{\frac{(1 - 1.5)^2+(2 - 1.5)^2}{2 - 1}}=\sqrt{0.25 + 0.25}=\sqrt{0.5}\approx0.707\)
• For sample \((1,12)\): \(\bar{x}=\frac{1+12}{2}=6.5\), \(s=\sqrt{\frac{(1 - 6.5)^2+(12 - 6.5)^2}{2-1}}=\sqrt{30.25+30.25}=\sqrt{60.5}\approx7.778\)
• For sample \((2,1)\): \(\bar{x}=\frac{2 + 1}{2}=1.5\), \(s=\sqrt{\frac{(2 - 1.5)^2+(1 - 1.5)^2}{2 - 1}}=\sqrt{0.25+0.25}=\sqrt{0.5}\approx0.707\)
• For sample \((2,2)\): \(\bar{x}=\frac{2+2}{2}=2\), \(s=\sqrt{\frac{(2 - 2)^2+(2 - 2)^2}{2-1}} = 0\)
• For sample \((2,12)\): \(\bar{x}=\frac{2+12}{2}=7\), \(s=\sqrt{\frac{(2 - 7)^2+(12 - 7)^2}{2-1}}=\sqrt{25 + 25}=\sqrt{50}\approx7.071\)
• For sample \((12,1)\): \(\bar{x}=\frac{12+1}{2}=6.5\), \(s=\sqrt{\frac{(12 - 6.5)^2+(1 - 6.5)^2}{2-1}}=\sqrt{30.25+30.25}=\sqrt{60.5}\approx7.778\)
• For sample \((12,2)\): \(\bar{x}=\frac{12+2}{2}=7\), \(s=\sqrt{\frac{(12 - 7)^2+(2 - 7)^2}{2-1}}=\sqrt{25+25}=\sqrt{50}\approx7.071\)
• For sample \((12,12)\): \(\bar{x}=\frac{12+12}{2}=12\), \(s=\sqrt{\frac{(12 - 12)^2+(12 - 12)^2}{2-1}} = 0\)

Step2: Create probability distribution table

The distinct \(s\) values are \(0\), \(0.707\), \(7.071\), \(7.778\)

• \(P(s = 0)=\frac{3}{9}=\frac{1}{3}\)
• \(P(s=0.707)=\frac{2}{9}\)
• \(P(s = 7.071)=\frac{2}{9}\)
• \(P(s=7.778)=\frac{2}{9}\)

Answer:

\(s\)	Probability
\(0.707\)	\(\frac{2}{9}\)
\(7.071\)	\(\frac{2}{9}\)
\(7.778\)	\(\frac{2}{9}\)