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  

1131

cc (cubic centimeters) and that the standard deviation is

299

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 533 cc and 1729.

(b) According to Chebyshev's theorem, at least ?56%75%84%89% of the measurements lie between 682.5 cc and 1579.5.

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the measurements lie between 533 cc and 1729.

(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

Ans:

mean=1131

standard deviation=299

a)

k=(533-1131)/299=-2 or (1729-1131)/299=+2

1-(1/k^2)=1-(1/2^2)=1-0.25=0.75,at least 75% of the measurements lie between 533 and 1729.

b)

k=(682.5-1131)/299=-1.5 or (1579.5-1131)/299=+1.5

1-(1/k^2)=1-(1/1.5^2)=0.56,at least 56% of the measurements lie between 682.5 and 1579.5.

c)533 and 1729 are 2 standard deviations below and above respectively.

So,According to the empirical rule, approximately 95% of the measurements lie between 533and 1729.

d)According to the empirical rule, approximately 68% of the measurements lie between within one standard deviation of the mean i.e. 1131-299=832 and 1131+299=1430