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 37 hours and a standard deviation of 5.5 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries

a.What can you say about the shape of the distribution of the sample mean?

b.What is the standard error of the distribution of the sample mean? (Round your answer to 4 decimal places.)

c.What proportion of the samples will have a mean useful life of more than 38.5 hours? (Round your z value to 2 decimal places and final answer to 4 decimal places.)

d)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.)

e) What proportion of the sample will have a mean useful life between 36.5 and 38.5 hours? (Round your z value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

Solution :

Given that ,

a) The shape of the distribution of the sample mean is normal

= 37

b) = / n = 5.5 / 9 = 1.8333

c)P( > 38.5) = 1 - P( < 38.5)

= 1 - P[( - ) / < (38.5 - 37) / 1.8333]

= 1 - P(z < 0.82)   

= 1 - 0.7939

= 0.2061

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

= 1 - P[( - ) / < (36.5 - 37) / 1.8333]

= 1 - P(z < -0.27)   

= 1 - 0.3936

= 0.6064

e) P(36.5 < < 38.5 )  

= P[(36.5 - 37)1.8333 / < ( - ) / < (38.5 - 37) / 1.8333 )]

= P( -0.27 < Z < 0.82)

= P(Z < 0.82 ) - P(Z < -0.27 )

Using z table,  

= 0.7939 - 0.3936

= 0.4003