Question

From the time of early studies by Sir Francis Galton in the late nineteenth century linking it with mental ability, the cranial capacity of the human skull has played an important role in arguments about IQ, racial differences, and evolution, sometimes with serious consequences. (See, for example, S.J. Gould, "The Mismeasure of Man," 1996.) Suppose that the mean cranial capacity measurement for modern, adult males is 1145 cc (cubic centimeters) and that the standard deviation is 262 cc. Complete the following statements about the distribution of cranial capacity measurements for modern, adult males. (a) According to Chebyshev's theorem, at least of the measurements lie between 752 cc and 1538 cc. (b) According to Chebyshev's theorem, at least of the measurements lie between 621 cc and 1669 cc. (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately of the measurements lie between 621 cc and 1669 cc. (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the measurements lie between cc and cc.

Answer 1

a)

since 752 cc and 1538 cc are 1.5 standard deviation from mean:

at least 55.56 % of the measurements lie between 752 cc and 1538 cc.

b)

since 621 cc and 1669 cc are 2 standard deviation from mean:

at least 75 % of the measurements lie between 621 cc and 1669 cc.

c)

According to the empirical rule, approximately 95% of the measurements lie between 621 cc and 1669 cc.

d)

since 68% of the values are 1 standard deviation from the mean value

According to the empirical rule, approximately 68% of the measurements lie between 883 and 1407 cc