Question

1. Assume the average number of years that nurses have worked at their job is normally distributed with a mean of 21 years and a population standard deviation of 12 years. Suppose we take a sample of 15 nurses.
   1. What is the probability that a randomly selected nurse will have worked for more than 30 years?
   2. What is the probability that a randomly selected nurse will have worked for under 5 years?
   3. What is the probability that a randomly selected nurse has worked between 10 and 25 years?
   4. What is the probability that the average number of years nurses have worked in the sample is above 25?
   5. What number of years worked defines the lowest 34.09% of the distribution for an individual nurse?
   6. What number of years worked defines the highest 7.93% of the distribution for an individual nurse?

Answer 1

Given that

μ= 21,σ = 12

a) Probability that a randomly selected nurse will have worked for more than 30 years

b) Probability that a randomly selected nurse will have worked for under 5 years

c) Probability that a randomly selected nurse has worked between 10 and 25 years

d)  Probability that the average number of years nurses have worked in the sample is above 25

The z-score for =25 is

z= x - μ/ σ / √n = 25 - 21 / 12 / √15 = 1.29

Using z table, the probability that the average number of years nurses have worked in the sample is above 25 is

P(X > 25)=1-P(z ≤ 1.29)=1-0.9015=0.0985

(e)

Here we need z-score that has 0.3409 area to its left.

The z-score -0.41 has 0.3409 area to its left. The required x is

-0.41 = x - μ/ σ = x - 21 / 12

x=16.08

(f)

Here we need z-score that has 1- 0.0793 = 0.9207 area to its left.

The z-score 1.42 has 0.9207 area to its left. The required x is

1.42= x - μ/ σ = x - 21 / 12

x=38.04