Question

The diameter of bushings turned out by a manufacturing process is a normally distributed random variable with a mean of 4.025 mm and a standard deviation of 0.105 mm. A sample of 42 bushings is taken once an hour. Within what interval should 95 percent of the bushing diameters fall?

Answer 1

Solution:

Given:The diameter of bushings turned out by a manufacturing process is a normally distributed random variable with a mean of 4.025 mm and a standard deviation of 0.105 mm.

Sample size = n = 42

We have to find an interval in which 95% of the bushing diameters fall.

That is find x1 and x2 such that:

P( x1 < X < x2 )= 95%

P( x1 < X < x2 )= 0.95

thus find z values such that:

P( -z < Z < z) = 0.95

Since 0.95 is middle area , find an area = ( 1+0.95) / 2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : z  = 1.96

thus -z = -1.96

Now use following formula to find x value:

and

Thus we get:

P( x1 < X < x2 ) = 95%

P( 3.8192 < X < 4.2308) = 95%

That is: 95% of the bushing diameters fall within ( 3.8192 , 4.2308 )