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 10, 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 1102 cc (cubic centimeters) and that the standard deviation is 289 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 le between 524 cc and 1680 CC. (b) According to Chebyshev's theorem, at least 36% of the measurements lie between Icc and C. (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 ct and cc. ? (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 2 of the measurements lie between 524 cc and 1680 CC

Answer 1

We are given here that:
Mean = 1102 and SD = 289

a) We have here:
524 = 1102 - 2*289 that is 2 standard deviations less than the mean.
1680 = 1102 + 2*289 that is 2 standard deviations more than the mean.

According to chebyshev's inequality theorem, at least 1 - 1/k² of the observations lies within k standard deviations of the mean. Therefore 1 - 1/2² = 0.75 that is 75% is the correct answer here.

b) 1 - 1/k² = 0.36

1/k² = 1 - 0.36

1/k² = 0.64

1/k = 0.8

k = 1/0.8 = 1.25

Therefore the range here is computed as:
L = 1102 - 289*1.25 = 740.75 cc
U = 1102 + 289*1.25 = 1463.25 cc

c) According to empirical rule, approx. 68% of the observations lies within 1 standard deviation of the mean. Therefore the limits here are computed as:
L = 1102 - 289* = 813 cc
U = 1102 + 289 = 1391 cc

d) According to empirical rule, 95% of the observations lies within 2 standard deviations of the mean. Therefore 95% is the correct answer here.