having problems with practice

Question

According to Forbes magazine, the 2023 average starting salary for new college graduates in humanities was $50,681 The average starting salary for new college graduates in engineering was $73,922 (Forbes.com website). Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in humanities is $14,000 . Assume that the standard deviation for starting salaries for new college graduates in engineering is $19,000 .

a. What is the probability that a new college graduate in engineering will earn a starting salary of at least $80,000 (to 4 decimals)?

 

b. What is the probability that a new college graduate in humanities will earn a starting salary of at least 80,000 (to 4 decimals)?

 

c. What is the probability that a new college graduate in humanities will earn a starting salary less than 50,000 (to 4 decimals)?

 

d. How much would a new college graduate in engineering have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in humanities (to 0 decimal)?

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What percentage of engineering students are expected to earn a starting salary higher than 99% of all starting salaries of humanities graduates (to 2 decimals)?

Answer 1

a. We are given the following:

1. Mean starting salary for engineering graduates (μ = $73,922)
2. Standard deviation for engineering salaries (σ = $19,000)
3. We are asked to find the probability that a salary is greater than or equal to $80,000.

We can standardize the value using the Z-score formula:

1. z = (X - μ) / σ
2. z = (80,000 - 73,922) / 19,000 = 0.3199

Using a calculator or the z-table, find the area to the right of the z-score 0.3199. The probability is 37.45%.

b. For humanities graduates, we are given:

1. Mean starting salary for humanities graduates (μ = $50,681)
2. Standard deviation for humanities salaries (σ = $14,000)
3. We are asked to find the probability that a salary is greater than or equal to $80,000.

Use a similar method to part (a) to find z-score for X = $80,000, and then find the area to the right. Probability is 1.81%

c. Using a similar method to part (b), find the z-score for X = $50,000 and find the area to the left. Probability is 48.06%.

d. The z-score for the 99th percentile is the z-score where the area to the left is 0.99. Find this on the z-table/calculator, and you get z = 2.326.

Solve for X, which is the salary a new college grad in engineering would need to make to have a starting salary higher than 99% of humanities graduates.

2.326 = (X - 50,681) / 14,000

X = $83,250

Calculate z-score for $83,250 under the distribution of new engineering college grads' starting salaries and then find the area to the right to find out what percentage of engineering students are expected to earn more than this. The percentage is 31.17%

Hope this was helpful.