69. The Russian inathematician Tchebysheff proved that for any real number > 1, the pro- portion of the measurements that fall more than & standard deviations from the mean can be no greater than i 'k Moreover, this holds for any disinbution. not just bell-shaped oncs

a) Find the upper bound for the proportion of measurements falling (i) more than two standard deviations from the mean, (ii) more than three standard deviations from the mean. (iii) more than ten standard deviations from the mean.

b) Compare the upper bound for k = 2 to the approximate proportion falling more than two standard deviations from the mean in a bell-shaped disinbution Why is there a dif- ference?