Question

Teacher salaries for a particular district are known to have a normal distribution

with a mean of $38,500 and a standard deviation of $880.

a)What is the probability that a randomly chosen teacher from this district makes less than $41,000?

b)What is the probability that a randomly chosen teacher from this district makes more than $37,000?

c)What is the probability that a randomly chosen teacher from this district has a salary between $36,000 and $38,000.

d)One veteran teacher comments that her salary is at the 95th percentile for the district. What is her salary?

Answer 1

Solution :

Given that,

mean = = 38,500

standard deviation = = 880

a ) P( x < 41,000 )

P ( x - / ) < ( 41,000 - 38,500 / 880)

P ( z < 2500 / 880 )

P ( z < 2.84)

= 0.9977

Probability =0.9977

b ) P (x > 37,000 )

= 1 - P (x < 37,000 )

= 1 - P ( x -  / ) < ( 37,000- 38,500 / 880)

= 1 - P ( z <- 1500 / 880 )

= 1 - P ( z < -1.70)

Using z table

= 1 - 0.0446

= 0.9554

Probability = 0.9554

c ) P (36,000 < x < 38,000 )

P ( 36,000 - 38,500 / 880) < ( x -  / ) < ( 38,000 - 38,500 / 880)

P ( - 2500 / 880 < z < -500 / 880 )

P (-2.84 < z < - 0.57)

P ( z < - 0.57 ) - P ( z < -2.84)

Using z table

= 0.2843 - 0.0023

= 0.2821

Probability = 0.2821

d ) P(Z < z) = 95%

= P(Z < z) = 0.95  

= P(Z < -0.6745 ) = 0.95

z = 1.64

Using z-score formula,

x = z * +

x = 1.64 * 880 + 38,500

x = 39943.2