1. (a) Of a total of N numbers, the fraction p are 1’s, while the fraction q = 1 - p are 0’s. Prove that the standard deviation of the set of numbers is √pq. (b) Apply the result of part (a) to...

1. (a) Of a total of N numbers, the fraction p are 1’s, while the fraction q = 1 - p are 0’s. Prove that the standard deviation of the set of numbers is √pq.

(b) Apply the result of part (a) to Problem 4.56(c).

2. (a) Prove that the variance of the set of n numbers a, a þ d, a þ 2d, . . . , a + (n – 1)d (i.e., an arithmetic progression with the first term a and common difference d) is given by 1/12 (n² - 1)d²

3. What empirical relation would you expect to exist between the semi-interquartile range and the mean deviation for bell-shaped distributions that are moderately skewed?

4. A frequency distribution that is approximately normal has a semi-interquartile range equal to 10. What values would you expect for (a) the standard deviation and (b) the mean deviation?