Three randomly selected households are surveyed. The numbers of people in the ho

Question

Three randomly selected households are surveyed. The numbers of people in the households are 4, 5, and 9. Assume that samples of size n -2 are randomly selected with replacement from the population of 4, 5, and 9. Listed below are the nine different samples. Complete parts (a) through (c) 4,4 4,5 4,9 5,4 5,5 5,9 9,4 9,5 9,9 a. Find the variance of each of the nine samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution of the distinct variance values. Probability (lype an integer or a fraction. Use ascending order of the sample variances.) b. Compare the population variance to the mean of the sample variances. Choose the correct answer below. A. The population variance is equal to the mean of the sample variances. O B. The population variance is equal to the square root of the mean of the sample variances ° C. The population variance is equal to the square of the mean of the sample variances c. Do the sample variances target the value of the population variance? In general, do sample variances make good estimators of population variances? Why or why not?

Explanation / Answer

(a) Here varaince of the given 9 samples.

Probability distribution of variances are

(b) Here mean of the sample variances = (0 * 3 + 0.5 * 2 + 8 * 2 + 12.5 * 2)/9 = 4.6667

Here population mean = (4 + 5 + 9)/3 = 6

Population variance = (6 - 4)2 * 1/3 + (6 - 5)2 * 1/3 + (6 - 9)2 * 1/3 = 14/3 = 4.6667

So option A is correct.

c. Yes sample varianes target the population variances. In general, sample variances make good estimators of population variances when there is high sample size.

sample (n1) sample(n2) Variance 4 4 0 4 5 0.5 4 9 12.5 5 4 0.5 5 5 0 5 9 8 9 4 12.5 9 5 8 9 9 0

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