Question

Power +, 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 5.4 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries. What can you say about the shape of the distribution of the sample mean? What is the standard error of the distribution of the sample mean? (Round your answer to 4 decimal places.) What proportion of the samples will have a mean useful life of more than 39 hours? (Round your z value to 2 decimal places and final answer to 4 decimal places.) What proportion of the sample will have a mean useful life greater than 36.5 hours? (Round your z value to 2 decimal places and final answer to 4 decimal places.) What proportion of the sample will have a mean useful life between 36.5 and 39 hours? (Round your z value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

Solution :

Given that ,

mean = = 38

standard deviation = = 5.4

n = 9

=   = 38

= / n = 5.4 / 9 = 1.8

a) P( > 39) = 1 - P( < 39)

= 1 - P[( - ) / < (39 - 38) / 1.8]

= 1 - P(z < 0.56)

Using z table,    

= 1 - 0.7123

= 0.2877

b) P( > 36.5) = 1 - P( < 36.5)

= 1 - P[( - ) / < (36.5 - 38) / 1.8]

= 1 - P(z < -0.83)

Using z table,    

= 1 - 0.2033

= 0.7967

c) P(36.5 < < 39)  

= P[(36.5 -38) /1.8 < ( - ) / < (39 -38) /1.8 )]

= P(-0.83 < Z < 0.56)

= P(Z < 0.56) - P(Z < -0.83)

Using z table,  

= 0.7123 - 0.2033

= 0.5090