Question

The quality control department of John Deere measured the length of 100 bolts randomly selected from a specified order. The mean length was found to be 9.75 cm, and the standard deviation was 0.01 cm. if the bolt lengths are normally distributed, find:

a) The percentage of bolts shorter than 9.74 cm

b) The percentage of bolts longer than 9.78 cm

c) The percentage of bolts that meet the length specification of 9.75 +/- 0.02 cm

d) The percentage of bolts that are longer than the nominal length o f9.75 cm

show work

Answer 1

Let x = The length of bolt

Given that x is normally distributed with

Mean

SD

Sample size = n = 100

A) we want to calculate percentage of bolts shorter than 9.74 cm i.e.

By using Z score we can calcute it

Where,   

For x = 9.74 ,   

i.e. we want to calculate

....................( From Z table)

Therefor, 15.86 % of bolts shorter than 9.74 cm ..................( ANSWER)

=========================================================================

B ) we want to calculate percentage of bolts longer than 9.78 cm i.e.

By using Z score we can calcute it

Where,   

For x = 9.78 ,   

i.e. we want to calculate

....................( From Z table)

Therefor, 0.13 % of percentage of bolts longer than 9.78 cm ..................( ANSWER)

=============================================================================

c) The percentage of bolts that meet the length specification of 9.75 +/- 0.02 cm.

i.e. Lower specification limit = 9.75 - 0.02 = 9.73

  Upper specification limit = 9.75 + 0.02 = 9.77

i.e we want to calculate percentage of bolts that meet the length specification i.e.

By using Z score we can calcute it

Where,   

For x = 9.73 ,   

For x = 9.77 ,   

i.e. we want to calculate

..................( By using Z table)

   .........................( ANSWER )

  95.45% Percentage of bolts that meet the length specification  ...........( ANSWER )

=================================================================================

d) The percentage of bolts that are longer than the nominal length of 9.75 cm i.e.

By using Z score we can calcute it

Where,   

For x = 9.75 ,   

i.e. we want to calculate

   ..................( From Z table)

   .....( ANSWER)

Therefor, 50% of bolts that are longer than the nominal length of 9.75 cm ..................( ANSWER)