Question

1. As reported in Runner’s World magazine, the times of the finishers in the New York City 10-km run are Normally distributed with mean 61 minutes and standard deviation 9 minutes.
   a. What percentage of runners finish with a time less than 75 minutes?
   b. What percentage of runners finish with a time between 50 and 70 minutes?

   c. What percentage of runners take longer than 90 minutes to finish?

2. The annual salaries or employees in a large company are approximately Normally distributed with a mean value of $50 thousand and a standard deviation of $20 thousand. a. What percentage of employees earn less than $40 thousand? b. What percentage of employees earn between $45 and $65 thousand? c. What percentage of employees earn more than $70,000? d. If an employee is in the bottom 2% of all salaries, what must his/her salary be? e. What would an employee have to earn to be in the top 1% of all salaries?

Answer 1

Solution :

Given that ,

mean = = 61

standard deviation = = 9

a). What percentage of runners finish with a time less than 75 minutes?

Ans: P(X<75) = P[(X- ) / < (75-61) / 9]

= P(z <1.55 )

Using z table

= 0.94009

=94.01%

b. What percentage of runners finish with a time between 50 and 70 minutes?

Ans:

P(50< x < 70) = P[(50- 61) / 9< (x - ) / < (70-61) / 9)]

= P(-1.22 < Z <1 )

= P(Z <1 ) - P(Z < -1.22)

Using z table   

= 0.8413-0.1112

=0.7301

=73.01%

c. What percentage of runners take longer than 90 minutes to finish?

Ans:P(X>90)=1-P(X<90)

P(X>90) =1- P[(X- ) / < (90-61) / 9]

= 1-P(z <3.22 )

Using z table

=1- 0.9994

=0.0006

=0.06%

#Solution b

mean = = 50

standard deviation = = 20

a. What percentage of employees earn less than $40 thousand?

Ans: P(X<40) = P[(X- ) / < (40-50) / 20]

= P(z <-0.5 )

Using z table

= 0.3085

=30.85%

b. What percentage of employees earn between $45 and $65 thousand?

Ans:P(45< x < 65) = P[ (45-50) / 20< (x - ) / < (65-50) / 20)]

= P(-0.25 < Z <0.75 )

= P(Z <0.75 ) - P(Z < -0.25)

Using z table   

= 0.7734-0.4013

=0.3721

=37.21%

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