Three randomly selected children are surveyed. The ages of the children are 3, 5 , and 10. Assume that samples of size $n=2$ are randomly selected with replacement from the population of 3,5 , and 10. Listed below are the nine different samples. Complete parts (a) through (d)
a. Find the value of the population variance o $\sigma^{2}$
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Answer
Final Answer: The population variance \(\sigma^{2}\) is approximately \(\boxed{8.67}\).
Step 1 ：The population consists of the ages of the three children: 3, 5, and 10.
Step 2 ：First, we need to calculate the mean (average) of the population.
Step 3 ：The mean of the population is \(\frac{3+5+10}{3} = 6.0\).
Step 4 ：Then, for each number in the population, we subtract the mean and square the result.
Step 5 ：The squared differences from the mean are \((3-6)^2 = 9\), \((5-6)^2 = 1\), and \((10-6)^2 = 16\).
Step 6 ：Finally, we calculate the average of these squared differences to get the population variance.
Step 7 ：The population variance is \(\frac{9+1+16}{3} = 8.666666666666666\).
Step 8 ：The population variance, calculated as the average of the squared differences from the mean, is approximately 8.67.
Step 9 ：Final Answer: The population variance \(\sigma^{2}\) is approximately \(\boxed{8.67}\).
