Example 3.10 Duracell manufactures AA batteries for use in a wide variety of com

Question

Example 3.10 Duracell manufactures AA batteries for use in a wide variety of commercial products. The production line of Duracell is designed and operated in such a way that X, the voltage of any given battery produced on the line, is a normal random variable having a mean of 1.5 V and a standard deviation of 0.01 V Sketch the probability density function of X. What proportion of batteries has a voltage of 1.5 V or less? What proportion of batteries has a voltage of 1.515 V or less? What is the probability that the voltage of a produced battery exceeds 1.49 V? What is the probability that the voltage of a produced battery is at least 1.48 V but no more than 1.52 V? Find the battery voltage x such that 2.5% of the batteries produced have a voltage greater than x. Find the battery voltage x such that 1% of the batteries produced have a voltage less than x. a. b. c. d. e. f. g. 46

Explanation / Answer

Ans:

Given that

mean=1.5

standard deviation=0.01

z=(x-1.5)/0.01

b)

z=(1.5-1.5)/0.01=0

P(x<=1.5)=P(z<=0)=0.5

c)

z=(1.515-1.5)/0.01=1.5

P(x<=1.5150=P(z<=1.5)=0.9332

d)

z=(1.49-1.5)/0.01=-1

P(x>1.49)=P(z>-1)=0.8413

e)

z(1.48)=(1.48-1.5)/0.01=-2

z(1.52)=(1.52-1.5)/0.01=2

P(-2<=z<2)=P(z<=2)-P(z<-2)=0.9772-0.0228=0.9545

f)P(Z>=z)=0.025

P(Z<=z)=0.975

z=1.96

x=1.5+1.96*0.01=1.5196

g)P(Z<=z)=0.01

z=-2.33

x=1.5-2.33*0.01=1.4767

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