Question

The number of letters handled daily by a post office is normally distributed with a mean of 20,000 letters and a standard deviation of 100 letters. Find the percent of days on which the post office handles:

A.) more than 20,200 letters

B.) fewer than 19,900 letters

C.) Between 19,950 and 20,050 letters

D.) between 20,100 and 20,150 letters

Answer 1

Solution:

Given that,

mean =   = 20,000  letters

standard deviation = = 100 letters

A ) P ( x > 20,200 )

= 1 - P (x < 12.4 )

= 1 - P ( x -  / ) < ( 20,200 - 20,000 / 100)

= 1 - P ( z < 200 / 100 )

= 1 - P ( z < 2 )

Using z table

= 1 - 0.9772

= 0.0228

Probability = 0.0228

B ) P( x < 19900 )

P ( x -  / ) < ( 19900 - 20000 / 100)

P ( z < - 100 / 100 )

P ( z < - 1 )

Using z table

= 0.1587

Probability = 0.1587

C ) P( 19950 < x < 20050 )

P ( 19950 - 20000 / 100) < ( x -  / ) < ( 20050 - 20000 / 100)

P ( < - 50 /100z < 50 / 100 )

P (- 0.5 < z < 0.5 )

P ( z < 0.5 ) -  P ( z < - 0.5 )

Using z table

= 0.6915 - 0.3085

= 0.3830

Probability = 0.3830

D ) P( 20100 < x < 20150 )

P ( 20100 - 20000 / 100) < ( x -  / ) < ( 20150 - 20000 / 100)

P ( < 100 /100z < 150 / 100 )

P ( 1 < z < 1.5 )

P ( z < 1.5 ) -  P ( z < 1 )

Using z table

= 0.9332 - 0.8413

= 0.0919

Probability = 0.0919