Answer
a) 0, 4/16
24.5, 2/16
2, 2/16
50, 2/16
40.5, 2/16
4.5, 2/16
72, 2/16
b) 24.2
c) Yes
Work Step by Step
a. We know there are 4 numbers, so there are $4^2=16$ ways to combine them. Combining the values with the same sample variances, we obtain the following:
Sample variance, probability:
0, 4/16
24.5, 2/16
2, 2/16
50, 2/16
40.5, 2/16
4.5, 2/16
72, 2/16
b. We know the following equation for sample variance: $\sigma^2 = \frac{\Sigma(x-\bar{x})^2}{n}$. Using this as well as the population variance, we see that the sample and population variances are both 24.2. (To verify, it is the easiest to use a program such as Microsoft Excel.)
c. The population variance is unbiased, for it is equal to the sample variance.
