Let X be the head-injury rating: X∼N(μ,σ2). Then Z=σX−μ∼N(0,1)
Given N=605,σ=185.
a.
P(500<X<700)=P(700)−P(X≤500)=
=P(Z<185700−605)−P(Z≤185500−605)≈
≈P(Z<0.513514)−P(Z≤−0.567568)≈
≈0.6962−0.2852=0.4110 b.
P(400<X<500)=P(500)−P(X≤400)=
=P(Z<185500−605)−P(Z≤185400−605)≈
≈P(Z<−0.567568)−P(Z≤−1.108108)≈
≈0.2852−0.1339=0.1513 c.
P(X<850)=P(Z<185850−605)≈
≈P(Z<1.324324)≈0.9073 d.
P(X>1000)=1−P(X≤1000)=
=1−P(Z<1851000−605)≈1−P(Z≤2.135135)≈
≈1−0.983625≈0.0164 e.
Pr(Z<zp)=0.1
zp≈−1.28155
P10=μ+zp×σ
P10=605+(−1.28155)×185≈368 f.
Pr(Z<zp)=0.95
zp≈1.645
P95=μ+zp×σ
P95=605+1.645×185≈909
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