Questions: Three randomly selected children are surveyed. The ages of the children are 1, 4, and 10. Assume that samples of size n=2 are randomly selected with replacement from the population of 1,4 , and 10. Listed below are the nine different samples. Complete parts (a) through (d). 1,1 1,4 1,10 4,1 4,4 4,10 10,1 10,4 10,10 Ine proportion is 0.333 . (Round to three decimal places as needed.) b. Find the proportion of odd numbers of each of the nine samples, then summarize the sampling distribution of the sample proportion of odd numbers in the format of a table representing the probability distribution of the distinct proportion values. Sample Proportion Probability 0 4/9 0.5 4/9 1 1/9 (Type integers or simplified fractions.) c. Find the mean of the sampling distribution of the sample proportion of odd numbers. The mean is (Round to three decimal places as needed.)

Solution Steps

The proportions of odd numbers in each of the nine samples are calculated as follows:

• Sample \( (1, 1) \): \( \frac{2}{2} = 1.0 \)
• Sample \( (1, 4) \): \( \frac{1}{2} = 0.5 \)
• Sample \( (1, 10) \): \( \frac{1}{2} = 0.5 \)
• Sample \( (4, 1) \): \( \frac{1}{2} = 0.5 \)
• Sample \( (4, 4) \): \( \frac{0}{2} = 0.0 \)
• Sample \( (4, 10) \): \( \frac{0}{2} = 0.0 \)
• Sample \( (10, 1) \): \( \frac{1}{2} = 0.5 \)
• Sample \( (10, 4) \): \( \frac{0}{2} = 0.0 \)
• Sample \( (10, 10) \): \( \frac{0}{2} = 0.0 \)

Thus, the proportions of odd numbers in each sample are: \[ [1.0, 0.5, 0.5, 0.5, 0.0, 0.0, 0.5, 0.0, 0.0] \]

Next, we summarize the sampling distribution of the sample proportions. The distinct proportions and their corresponding probabilities are:

• Proportion \( 1.0 \): Probability \( \frac{1}{9} \)
• Proportion \( 0.5 \): Probability \( \frac{4}{9} \)
• Proportion \( 0.0 \): Probability \( \frac{4}{9} \)

Thus, the probability distribution is: \[ \begin{array}{c|c} \text{Sample Proportion} & \text{Probability} \\ \hline 1.0 & \frac{1}{9} \\ 0.5 & \frac{4}{9} \\ 0.0 & \frac{4}{9} \\ \end{array} \]

The mean of the sampling distribution of the sample proportion of odd numbers is calculated as follows:

\[ \text{Mean} = 1.0 \times \frac{1}{9} + 0.5 \times \frac{4}{9} + 0.0 \times \frac{4}{9} = \frac{1}{9} + \frac{2}{9} + 0 = \frac{3}{9} = \frac{1}{3} \approx 0.333 \]

Final Answer