Question

efficient experts study the processes used to manufacture items in order to make them as efficient as possible. One of the steps used to manufacture a metal clamp involves the drilling of three holes. in a sample of 75 clamps, the mean time to complete this step was 49.4 seconds. assume that the population standard deviation is 7 seconds. construct a 99% confidence interval for the mean time needed to complete this step. Round the answer to at least one decimal place.

Part B of question A sample size of ______ is needed in order to obtain a 80% confidence level with a margin error of 1.5 Round the sample size up to the nearest integer

Answer 1

Solution:

Part A)

Given,

= 49.4   

= 7

n = 75   

Note that, Population standard deviation() is known..So we use z distribution. Our aim is to construct 99% confidence interval.

c = 0.99

= 1- c = 1- 0.99 = 0.01

  /2 = 0.01 2 = 0.005 and 1- /2 = 0.995

Search the probability 0.995 in the Z table and see corresponding z value

= 2.576   

The margin of error is given by

E =  /2 * ( / n )

= 2.576 * (7 / 75)

= 2.08

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(49.4 - 2.08)   <   <  (49.4 + 2.08)

47.32 <   < 51.48

Required 99% confidence interval is (47.32 , 51.48)

Part B)

Solution:

Given ,

= 7 ..Population SD

E = 1.5 Margin of error

c = 80% = 0.80 ...confidence level

Find sample size required.

c = 0.80

= 1- c = 1- 0.80 = 0.20

  /2 = 0.10

Using Z table ,

= 1.28

Now, sample size (n) is given by,

=  {(1.28* 7 )/ 1.5}²

=  35.6807111111

= 36 ..(round to the next whole number)

Answer :A sample size of 36 is needed .