Consider the data sets portrayed in Figs. 3.5 and 3.6 on page 108.
a. Chebychev’s rule says that at least 75% of the observations lie within two standard deviations to either side of the mean. What percentage of the observations portrayed in Fig. 3.5 actually lie within two standard deviations to either side of the mean?
b. Chebychev’s rule says that at least 89% of the observations lie within three standard deviations to either side of the mean. What percentage of the observations portrayed in Fig. 3.5 actually lie within three standard deviations to either side of the mean?
c. Repeat parts (a) and (b) for the data portrayed in Fig. 3.6.
d. From parts (a)–(c), we see that Chebychev’s rule provides a lower bound on, rather than a precise estimate for, the percentage of observations that lie within a specified number of standard deviations to either side of the mean. Nonetheless, Chebychev’s rule is quite important for several reasons. Can you think of some?