The length of a screw produced by a machine is normally distributed with a mean of 0.25 inches and a standard deviation of 0.01 inches. What percent of screws are between 0.24 and 0.26 inches?

1 Answer
Jul 27, 2016

Screws between 0.24 and 0.26 inches = 68.26%

Explanation:

Given -

Mean #barx = 0.25#
SD #sigma=0.01#

Values are normally distributed.

Look at the graph.

Mean is presented exactly at the middle along the X-axis.
The other two # x# values are 0.24 and 0.26

They are also represented along the X - axis.

Their corresponding #z# values are represented below the # x# values, using the formula #z=(x-mu)/sigma#

At #x=0.25; z= (0.25-0.25)/0.01=0/0.01=0#

At #x=0.26; z= (0.26-0.25)/0.01=0.01/0.01=1#

At #x=0.24; z= (0.24-0.25)/0.01=-0.01/0.01=-1#

Using the Area under Normal Distribution Table you have to find the area between #z=0 and z=1#

This is same for #z=0 and z=-1#

Area between #[z=-1 and z=1] = # area between # [z=0 and z=-1] + # area between# [z=0 and z=1]#

Area between #[z=-1 and z=1] = 0.3413 + 0.3413=0.6826#

If one screw is taken at random the probability of its length fall between 0.24 and 0.26 inches is #=0.6826#

Then percentage of screws fall between 0.24 and 0.26 inches is #=0.6826 xx 100=68.26#

Look at the diagram

Normal Distribution Part - 1

Normal Distribution Part - 2