Question

The ages (in years) of the four U.S. vice presidents who assumed office after presidential assassinations are 42 (T. Roosevelt), and 55 (L.B. Johnson). (i) Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. (ii) Find the standard deviation of each of the 16 samples, then summarize the sampling distribution of the standard deviation in the format of a table representing the probability distribution. (iii) Compare the population standard deviation to the mean of the sample standard deviations. (iv) Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not?

Answer 1

This detailed answer explains how to list the possible samples, calculate standard deviations, and compares population and sample standard deviations.

Step-by-step explanation:

(i) To find the 16 different possible samples when 2 of the ages are randomly selected with replacement, we can use a combination formula. The samples are:

42, 42
42, 55
55, 42
55, 55
42, 42
42, 55
55, 42
55, 55
42, 42
42, 55
55, 42
55, 55
42, 42
42, 55
55, 42
55, 55

(ii) To find the standard deviation of each of the 16 samples, we need to calculate the standard deviation using the formula:

standard deviation = sqrt((x1 - mean)^2 + (x2 - mean)^2)/2

After calculating the standard deviation for each sample, we can summarize the sampling distribution in a table representing the probability distribution.

(iii) The population standard deviation is a measure of the variability of the entire population, while the mean of the sample standard deviations gives an estimation of the average variability of all possible samples.

(iv) Sample standard deviations can be good estimators of population standard deviations if the samples are representative and there is no significant bias or error in the sampling process.