Question

Power t, Inc. produces AA batteries used in remote-controlled toy cars. The mean life of these batteries follows the normal probability distribution with a mean of 38 hours and a standard deviation of 5.9 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 16 batteries. a. What can you say about the shape of the distribution of the sample mean? normal b. What is the standard error of the distribution of the sample mean? (Round yohr answer to 4 decimal places.)

Answer 1

a) the distribution is "normal" because if population is normally distributed then sample is also normal distributed

b) standard error =σ/√n =5.9/√16 =5.9/4 =1.475

c)

μ=38

σ=5.9

x=39

z=(x-μ)/(σ/√n)

=(39-38)/(5.9/√16)

=1/1.475

=0.68

P(x>39) =P(z>0.68)

               =1-P(z<0.68)

               =1-0.7517

               =0.2483

d)

μ=38

σ=5.9

x=36.5

z=(x-μ)/(σ/√n)

=(36.5-38)/(5.9/√16)

=-1.5/1.475

=-1.02

P(x>36.5) =P(z>-1.02)

               =1-P(z<-1.02)

               =1-P(x>39) =P(z>0.68)

               =1-P(z<0.68)

               =1-0.1539

               =0.8461

e)

μ=38

σ=5.9

For x=36.5

z=(x-μ)/(σ/√n)

=(36.5-38)/(5.9/√16)

=-1.5/1.475

=-1.02

For x=39

z=(x-μ)/(σ/√n)

=(39-38)/(5.9/√16)

=1/1.475

=0.68

P(36.5<x<39) =P(-1.02<z<0.68)

                              =P(z<0.68)-P(z<-1.02)

                              =0.7517-0.1539

                              =0.5978