Question

A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a​ 99% confidence interval for the mean length cut by machine. Determine whether the statement is true or false. The confidence interval indicates that the machine is not working properly.

Answer 1

Solution:

Given,

n = 70 ....... Sample size

= 12.14 ....... Sample mean

= 0.15 ........population standard deviation

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 * (0.15 / 70 )

= 0.0462

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

( - E ) <   <  ( + E)

(12.14   - 0.0462)   <   <  (12.14  + 0.0462)

12.0938   <   <  12.1862

Required interval is (12.0938 , 12.1862)

Observe that , the interval does not contain the value 12 which is the desired mean length of the insulation . So , machine is not working properly.

Statement is TRUE