Question

Suppose that, on average, electricians earn approximately μ = 54, 000 dollars per year in the United States. Assume that the distribution for electricians' yearly earnings is normally distributed and that the standard deviation is σ = 12, 000.

1. (3 points) What is the probability that a randomly selected electrician's salary is more than $50,000 but less than $60,000?

2. (3 points) Tax code dictates that 40% income tax is applied to the electricians whose annual income is among top 5% in the population. What is the minimum yearly earnings that will be levied by 40% tax, i.e., ?nd the constant a such that Pr{X > a} = 5%.

3. (3 points) In a sample of four electricians, what is the probability that all four electricians' salaries are more than $50,000 but less than $60,000? (Please round your answer to 4 decimal places.)

4. (3 points) In a sample of four electricians, let X ̄ be the average yearly earnings of these four electricians. What is the expectation of the average earnings, i.e., E(X ̄) =?

5. (3 points) In a sample of four electricians, let X ̄ be the average yearly earnings of these four 9electricians. What is the standard deviation of the average earnings, i.e., s.e.(X ̄ ) =?

6. (3 points) Is the average earnings among four electricians, X ̄, normally distributed or not? Please explain. (No credit if there is no explanation.)

7. (3 points) What is the probability that the average salary of four randomly selected electricians is more than $50,000 but less than $60,000, i,e, Pr{50, 000 < X ̄ < 60, 000}? (If you are able to ?nd this probability, please show your answer; if you are not able to ?nd this probability, please explain why you can not.)

8. (4 points) What is the probability that the average salary of sixteen randomly selected electricians is more than $50,000 but less than $60,000 ? (If you are able to ?nd this probability, please show your answer; if you are not able to ?nd this probability, please explain why you can not.)

Answer 1

Let the random variable X is defined as

X: Electricians yearly earning

1) Required Probability = P ( 50000 < X < 60000)

= P ( -0.3333 < Z < 0.5000), since Z = (X- E(X)) / SD(X) ~ N(0,1)

= P ( Z< 0.5000) - P(Z< -0.3333)

From normal probability table

P(Z< 0.5000) = 0.6915 and P(Z< -0.3333) = 0.3695

P(50000< X < 60000) = 0.6915-0.3695 = 0.3220

2) P(X> a) = 0.05

---------------(I)

from normal probability table

P ( Z> 1.645) = 0.05-----------(II)

from (I) and (II)

(a-54000) / 12000 = 1.645

a= 73740

Minimum early income = $ 73740

3) n = 4

Since each electricians are independent to each other and probability remains constant for each electricians

P(50000< X< 60000) = 0.3220

P ( All four electricians salaries more than 50000 but less than 60000) = 0.3220⁴ = 0.0108

4) : Average income of 4 electricians

5) Variance of average salary of 4 electricians

Hence standard deviation of average salary of 4 electricians

6) If X is normal variable, then the distribution of sampling mean is also standard normal.

The distribution of sampling is

7) Required probability = P ( 50000 < Xbar < 60000)

= P ( -0.6667 < Z < 1)

= P ( Z< 1) - P(Z< -0.6667)

from normal probability table

P(Z<1) = 0.8413 and P( Z< -0.6667) = 0.2525

P ( 50000 < Xbar < 60000) = 0.8413 -0.2513 = 0.5888

8) n = 16

The distribution of sampling is

Required probability = P ( 50000 < Xbar < 60000)

= P ( -1.3333 < Z < 2)

= P ( Z< 2) - P(Z< -1.3333)

from normal probability table

P(Z<-1.3333) =0.0913 and P( Z< 2) = 0.9772

P ( 50000 < Xbar < 60000) = 0.9772-0.0913= 0.8859.