Question

Suppose the life of a particular brand of calculator battery is approximately normally distributed with a mean of 80 hours and a standard deviation of 9 hours. Complete parts a through c.

a. What is the probability that a single battery randomly selected from the population will have a life between 7575 and 85 ​hours?

b. What is the probability that 99 randomly sampled batteries from the population will have a sample mean life of between 75 and 85 ​hours?

c. If the manufacturer of the battery is able to reduce the standard deviation of battery life from 9 to 7 ​hours, what would be the probability that 9 batteries randomly sampled from the population will have a sample mean life of between 75 and 85 ​hours?

Answer 1

the pdf of normal distribution is = 1/σ * √2π * e ^ -(x-u)^2/ 2σ^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( z )= x - u / sd ~ n(0,1)
mean ( u ) = 80
standard deviation ( sd )= 9
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(a) to find p(a < = z < = b) = f(b) - f(a)
p(x < 75) = (75-80)/9
= -5/9 = -0.5556
= p ( z <-0.5556) from standard normal table
= 0.2893
p(x < 85) = (85-80)/9
= 5/9 = 0.5556
= p ( z <0.5556) from standard normal table
= 0.7107
p(75 < x < 85) = 0.7107-0.2893 = 0.4215
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(b) when sampled sample size (n) = 9
mean of the sampling distribution ( x ) = 80
standard deviation ( sd )= 9/ Sqrt ( 9 ) =3
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 75) = (75-80)/9/ Sqrt ( 9 )
= -5/3
= -1.6667
= P ( Z <-1.6667) From Standard Normal Table
= 0.0478
P(X < 85) = (85-80)/9/ Sqrt ( 9 )
= 5/3 = 1.6667
= P ( Z <1.6667) From Standard Normal Table
= 0.9522
P(75 < X < 85) = 0.9522-0.0478 = 0.9044
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(c) for sample size (n) = 9, sample standard deviation = 7
standard deviation ( sd )= 7/ Sqrt ( 9 ) =2.3333
to find p(a <= z <=b) = f(b) - f(a)
p(x < 75) = (75-80)/7/ sqrt ( 9 )
= -5/2.3333
= -2.1429
= p ( z <-2.1429) from standard normal table
= 0.0161
p(x < 85) = (85-80)/7/ sqrt ( 9 )
= 5/2.3333 = 2.1429
= p ( z <2.1429) from standard normal table
= 0.9839
p(75 < x < 85) = 0.9839-0.0161 = 0.9679
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