Question

three randomly selected children are surveyed. the ages of the children are 2, 7, and 12. assume that samples of size ( n = 2 ) are randomly selected with replacement from the population of 2, 7, and 12. listed below are the nine different samples. complete parts (a) through (d).
2,2 2,7 2,12 7,2 7,7 7,12 12,2 12,7 12,12

a. for the population, find the proportion of even numbers.
the proportion is 0.667
(round to three decimal places as needed.)
b. find the proportion of even numbers of each of the nine samples, then summarize the sampling distribution of the sample proportion of even numbers in the format of a table representing the probability distribution of the distinct proportion values.

sample
proportion
probability

(type integers or simplified fractions.)

Explanation:

Step1: Calculate sample proportion for each sample

• For sample \(2,2\): Proportion \(=\frac{2}{2} = 1\)
• For sample \(2,7\): Proportion \(=\frac{1}{2}=0.5\)
• For sample \(2,12\): Proportion \(=\frac{2}{2} = 1\)
• For sample \(7,2\): Proportion \(=\frac{1}{2}=0.5\)
• For sample \(7,7\): Proportion \(=\frac{0}{2}=0\)
• For sample \(7,12\): Proportion \(=\frac{1}{2}=0.5\)
• For sample \(12,2\): Proportion \(=\frac{2}{2} = 1\)
• For sample \(12,7\): Proportion \(=\frac{1}{2}=0.5\)
• For sample \(12,12\): Proportion \(=\frac{2}{2} = 1\)

Step2: Count the frequency of each proportion value

• Proportion \(0\): Frequency \(=1\)
• Proportion \(0.5\): Frequency \(=4\)
• Proportion \(1\): Frequency \(=4\)

Step3: Calculate probability for each proportion value

• Probability of \(0\): \(P(0)=\frac{1}{9}\)
• Probability of \(0.5\): \(P(0.5)=\frac{4}{9}\)
• Probability of \(1\): \(P(1)=\frac{4}{9}\)

Answer:

Sample Proportion	Probability
\(0.5\)	\(\frac{4}{9}\)
\(1\)	\(\frac{4}{9}\)