Questions: 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 22 2.12 12,1 12.2 12,12 a. Find the value of the population standard deviation σ. σ=4.987 (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. Standard Deviation Probability (Type integers or fractions.)

Solution Steps

To solve this problem, we need to follow these steps:

a. Calculate the population standard deviation. b. Calculate the standard deviation for each of the nine samples. c. Summarize the sampling distribution of the standard deviations in a table.

1. Calculate the population standard deviation using the given population values.
2. For each sample, calculate the sample standard deviation.
3. Create a table to represent the probability distribution of the distinct standard deviation values.

Given the population values \( \{1, 2, 12\} \), the population standard deviation is calculated as: \[ \sigma = 4.967 \]

For each of the nine samples, the standard deviations are: \[ \begin{align_} \{1, 1\} & : 0.000 \\ \{1, 2\} & : 0.500 \\ \{1, 12\} & : 5.500 \\ \{2, 1\} & : 0.500 \\ \{2, 2\} & : 0.000 \\ \{2, 12\} & : 5.000 \\ \{12, 1\} & : 5.500 \\ \{12, 2\} & : 5.000 \\ \{12, 12\} & : 0.000 \\ \end{align_} \]

The probability distribution of the distinct standard deviation values is: \[ \begin{array}{|c|c|} \hline \text{Standard Deviation} & \text{Probability} \\ \hline 0.000 & \frac{1}{3} \\ 0.500 & \frac{2}{9} \\ 5.000 & \frac{2}{9} \\ 5.500 & \frac{2}{9} \\ \hline \end{array} \]

Final Answer