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. hours and a standard deviation of 5.8 hours. As a part of its quality assurance program, Power +, Inc. tests samples of 9 batteries.

( I specifically would like to know how to get z step by step for C and D)

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

   

Sample mean

(Click to select)NormalUniformBinomial

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

   

Standard error

c.	What proportion of the samples will have a mean useful life of more than 39.5 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

   

Probability

d.	What proportion of the sample will have a mean useful life greater than 37.5 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

   

Probability

e.	What proportion of the sample will have a mean useful life between 37.5 and 39.5 hours? (Round z value to 2 decimal places and final answer to 4 decimal places.)

   

Probability

Answer 1

(a) The distribution of sample means is Normal Distribution.

(b) SE = /

        = 5.8/= 1.9333

(c) To find P( > 39.5):

Z = ( - )/SE

= (39.5 - 38)/1.9333 = 0.78

Table of Area Under Standard Normal Curve gives area = 0.2823

So,

P(>39.5) =0.5 - 0.2823 = 0.2177

(d)

To find P( > 37.5):

Z = ( - )/SE

= (37.5 - 38)/1.9333 = - 0.26

Table of Area Under Standard Normal Curve gives area = 0.1026

So,

P(>39.5) =0.5 + 0.1026 = 0.6026

(e)

To find P(37.5 < <39.5):

Case 1: For from 37.5 to mid value:

Z = ( - )/SE

= (37.5 - 38)/1.9333 = - 0.26

Table of Area Under Standard Normal Curve gives area = 0.1026

Case2 : For from mid value to 39.5:

Z = ( - )/SE

= (39.5 - 38)/1.9333 = 0.78

Table of Area Under Standard Normal Curve gives area = 0.2823

P(37.5 < < 39.5) =0.1026 + 0.2823 = 0.3849