PROBLEM 17.1 Let Z be a standard normal random variable. Calculate the following

Question

PROBLEM 17.1

Let Z be a standard normal random variable. Calculate the following probabilities and percentiles:

a.P(0 < Z < 2.14)

b.P(–1.32 < Z < 0.95)

c.P(Z > –1.18)

d.The 91st percentile of Z.

e.The 9th percentile of Z.

f.The 20th percentile of Z.

PROBLEM 17.2

A machines cuts corks intended for use in wine bottles. It produces corks with diameters that are normal with mean 3.06cm and standard deviation 0.08cm. Acceptable corks have diameters between 2.9cm and 3.1cm.

a.What % of the corks produced by the machine is acceptable?

b.84% of the corks produced by the machine have diameters less than _______?

Explanation / Answer

17.1.a.P(0<z<2.14)

=P(z<2.14)-P(z<0)

=0.9838-0.5

=0.4838

b.P(-1.32<z<0.95)

=P(z<0.95)-P(z<-1.32)

=0.8289-0.0934=0.7355

c.P(z>-1.18)

=1-P(z<-1.18)

=1-0.1190

=0.881

d.The z score corresponding to the area 0.91 is: 1.34

Thus the 91st percentile is 1.34

e.The z score corresponding to the area 0.09 is: -1.35.

Thus the 9th percentile is -1.35.

f. The z score corresponding to the area 0.20 is: -0.85.

Thus the 20th percentile is -0.85.

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