Suppose that the mean cranial capacity measurement for modern, adult males is 11

Question

Suppose that the mean cranial capacity measurement for modern, adult males is 1131cc (cubic centimeters) and that the standard deviation is 238cc. Complete the following statements about the distribution of cranial capacity measurements for modern, adult males.

(a) According to Chebyshev's theorem, at least 84% of the measurements lie between

cc

and

cc

. (Round your answer to the nearest integer.)

(b) According to Chebyshev's theorem, at least ?56%75%84%89% of the measurements lie between 655 cc and 1607 cc.

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the measurements lie between 655 cc and 1607 cc.

(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the measurements lie between

cc

and

cc

.

Explanation / Answer

let X be the random variable denoting cranial capacity measurement for modern, adult males

E[X]=u=1131 cc SD(X)=s=238 cc

a) Chebyshev's theorem tells us

P[|X-u|<st]>1-1/t2

or P[u-st<X<u+st]>1-1/t2

now we need to find that interval within which at least 84% of the measurements exist

so 1-1/t2=0.84

or 1/t2=0.16

or 1/t=0.4

or, t=2.5

hence the interval is u-2.5s and u+2.5s

now u=1131 cc     s=238 cc

so the interval is 1131-2.5*238=536cc and 1131+2.5*238=1726cc

hence

According to Chebyshev's theorem, at least 84% of the measurements lie between

536 cc and 1726 cc [answer]

b) here we have u-st=655 and u+st=1607

so 1131-238t=655 or t=2

so P[655<X<1607]=P[1131-2*238<X<1131+2*238]>1-1/22=0.75=75% [answer] [by chebyshev's theorem]

c) Suppose that the distribution is bell-shaped

which means X~N(1131,2382)

so P[655<X<1607]=P[(655-1131)/238<(X-1131)/238<(1607-1131)/238]=P[-2<Z<2] where Z~N(0,1)

=P[Z<2]-P[Z<-2]=0.977250-0.0227501=0.95=95% [answer]

d) let P[-a<Z<a]=0.997

or P[Z<a]-P[Z<-a]=0.997

or,2P[Z<a]-1=0.997   [as P[Z<-a]=1-P[Z<a]]

or P[Z<a]=0.9985=P[Z<2.96774]

so a=2.96774

now Z<2.96774

means (X-1131)/238<2.96774

or X<1837.322 cc

and -2.96774<Z

means -2.96774<(X-1131)/238

or 424.678<X

so According to the empirical rule, approximately 99.7% of the measurements lie between

424.678 cc and 1837.322 cc [answer]

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