Provide an appropriate response. 15) Three randomly selected households are surv

Question

Provide an appropriate response. 15) Three randomly selected households are surveyed as a pilot project for a larger survey to be co The numbers of people in the households are 4, 6, and 9. Consider the values of 4, 6, and 9 to be a population. Assume that samples of size n 2 are randomly selected The nine different samples are as follows: (4, 4), (4, 6), (4, 9), (6, 4), (6, 6), (6, 9), (9, 4), (9, 6), and (9, 9).(i) 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. (ii) Compare the population standard deviation to the mean of the sample standard deviations. (iii) Do the sample standard deviations target the value of the population standard deviation? In general, do standard deviations make good estimators of population standard deviation? Why on why not? with replacement from the population of 4, 6, and 9.

Explanation / Answer

(i) Standard deviation of the sample of size n = 2

Probability distribution of Sampling distribution of standard deviation

As frequncy of s = 0 is 3 and for rest of the s, it is 2.

E(s) = 0 * 0.3333 + 1.414 * 0.2222 + 2.121 * 0.2222 + 3.536 * 0.2222

= 1.5711

Population standard deviation () = sqrt [ {(4 - 6.333)2 + (6 - 6.333)2 + (9 - 6.333)2 }/3] = 2.0548

(iii) Yes, the sample standard deviations target the value of the population standard deviations. In general, for large size of population and sample, standard deviations are good estimators of population standard deviation. It is because sample is part of the population distribution.

Sample ID Sample 1 Sample2 Std. dev. 1 4 4 0 2 4 6 1.414 3 4 9 3.536 4 6 4 1.414 5 6 6 0 6 6 9 2.121 7 9 4 3.536 8 9 6 2.121 9 9 9 0

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