The U. S. Bureau of Economic Statistics reports that the average annual salary in the metropolitan Boston area is $50,542. Suppose annual salaries in the metropolitan Boston area are normally distributed with a standard deviation of $4,246. A Boston worker is randomly selected. (Round the values of z to 2 decimal places. Round your answers to 4 decimal places.) (a) What is the probability that the worker’s annual salary is more than $57,000? (b) What is the probability that the worker’s annual salary is less than $46,000? (c) What is the probability that the worker’s annual salary is more than $40,000? (d) What is the probability that the worker’s annual salary is between $45,000 and $54,000?

(a) P(X > $57,000) = 0.0643

(b) P(X < $46,000) = 0.1423

(c) P(X > $40,000) = 0.0066

(d) P($45,000 < X < $54,000) = 0.6959

Step-by-step explanation:

We are given that U.S. Bureau of Economic Statistics reports that the average annual salary in the metropolitan Boston area is $50,542.

Suppose annual salaries in the metropolitan Boston area are normally distributed with a standard deviation of $4,246.

Let X = annual salaries in the metropolitan Boston area

SO, X ~ Normal()

The z-score probability distribution for normal distribution is given by;

                      Z  =    ~ N(0,1)

where, = average annual salary in the Boston area = $50,542

             = standard deviation = $4,246

(a) Probability that the worker’s annual salary is more than $57,000 is given by = P(X > $57,000)

    P(X > $57,000) = P( > ) = P(Z > 1.52) = 1 - P(Z 1.52)

                                                                     = 1 - 0.93574 = 0.0643

The above probability is calculated by looking at the value of x = 1.52 in the z table which gave an area of 0.93574.

(b) Probability that the worker’s annual salary is less than $46,000 is given by = P(X < $46,000)

    P(X < $46,000) = P( < ) = P(Z < -1.07) = 1 - P(Z 1.07)

                                                                     = 1 - 0.85769 = 0.1423

The above probability is calculated by looking at the value of x = 1.07 in the z table which gave an area of 0.85769.

(c) Probability that the worker’s annual salary is more than $40,000 is given by = P(X > $40,000)

    P(X > $40,000) = P( > ) = P(Z > -2.48) = P(Z < 2.48)

                                                                     = 1 - 0.99343 = 0.0066

The above probability is calculated by looking at the value of x = 2.48 in the z table which gave an area of 0.99343.

(d) Probability that the worker’s annual salary is between $45,000 and $54,000 is given by = P($45,000 < X < $54,000)

    P($45,000 < X < $54,000) = P(X < $54,000) - P(X $45,000)

    P(X < $54,000) = P( < ) = P(Z < 0.81) = 0.79103

    P(X $45,000) = P( ) = P(Z -1.31) = 1 - P(Z < 1.31)

                                                                      = 1 - 0.90490 = 0.0951

The above probability is calculated by looking at the value of x = 0.81 and x = 1.31 in the z table which gave an area of 0.79103 and 0.9049 respectively.

Therefore, P($45,000 < X < $54,000) = 0.79103 - 0.0951 = 0.6959

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