13.12. (For this exercise and the following three, read the Note at the end of Chapter 13.) Use the table of normal deviates (Table 7, appendix) and obtain a sample of size 30 from a random variable X having a N(1, 4) distribution. Use this sample to answer the following:

(a) Compare P(X 2) with the relative frequency of that event.

(b) Compare the sample mean and the sample variance S2 with 1 and 4, respectively.

(c) Construct a graph of F(t) = P(X). Using the same coordinate system, obtain the graph of the empirical distribution function F,, defined as follows: F,(t) = 0 k/n if 130 from a random variable X having a N(1, 4) distribution. Use this sample to answer the following:

(a) Compare P(X 2) with the relative frequency of that event.

(b) Compare the sample mean X and the sample variance S2 with 1 and 4, respectively.

(c) Construct a graph of F(t) = P(X ). Using the same coordinate system, obtain the graph of the empirical distribution function F, defined as follows: F(t) = 0 = k/n = 1 if 1 < X(1) if if X(k) < < X(k+1) X(), where X is the ith largest observation in the sample (i.e., X is the ith order statistic). [The function F., is frequently used to approximate the cdf F. It may be shown that under fairly general conditions lim F(t) = F(t).]