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 1006 cc (cubic centimeters) and that the standard deviation is 240 cc.Complete the following statements about the distribution of cranial capacity measurements for modern, adult males. (a) According to Chebyshev's theorem, at least ?56%75%84%89% of the measurements lie between 526 cc and 1486. (b) According to Chebyshev's theorem, at least 8/9 (about 89%) of the measurements lie between cc and cc . (Round your answer to the nearest integer.) (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the measurements lie between cc and cc . (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the measurements lie between 526 cc and 1486.

Answer 1

Ans:

a)

526 and 1486 are 2 standard deviations below and above the mean respectively.

k=2

So,at least percent of the measurements lie between 526 cc and 1486

=1-(1/2^2)

=0.75 or 75%

b)

0.89=1-(1/k^2)

k=sqrt(1/0.11)

k=3.00

lower limit=1006-3*240=286

upper limit=1000+3*240=1726

c)68% of the data falls within one standard deviation.

lower limit=1006-240=766

upper limit=1006+240=1246

d)95% of data falls within two standard deviations.

526 and 1486 are 2 standard deviations below and above the mean respectively.

So,95% of the measurements lie between 526 cc and 1486.