The average salary for graduates entering the actuarial field is $40,000. If the salaries are normally distributed with a standard deviation of $5000 (σ = 5000), find the probability that a graduate will have a salary over $45,000: P(X > 45,000).

To find the probability (P) that a graduate will have a salary over $45,000, we need to calculate the z-score and then find the corresponding area under the standard normal distribution curve.

The formula to calculate the z-score is:

z = (x - μ) / σ

where x is the value we want to find the probability for, μ is the mean (average salary), and σ is the standard deviation.

In this case, x = $45,000, μ = $40,000, and σ = $5,000. So we can plug in these values to find the z-score:

z = (45,000 - 40,000) / 5,000
z = 5,000 / 5,000
z = 1

Now we need to find the corresponding area under the standard normal distribution curve for z = 1. We can use a table or a calculator to find this area. Let's assume we are using a standard normal distribution table.

The table provides the area to the left of a given z-score. Since we are looking for the probability that a graduate will have a salary over $45,000 (P(X > 45,000)), which is the area to the right of z = 1, we need to find the complement of the area to the left of z = 1.

The complement of an area A is equal to 1 - A. Therefore, we need to find 1 - the area to the left of z = 1.

Looking up the z-score of 1 in the table, we find that the area to the left of z = 1 is approximately 0.8413. Therefore, the complement is:

1 - 0.8413 = 0.1587

So the probability that a graduate will have a salary over $45,000 (P(X > 45,000)) is approximately 0.1587, or 15.87%.

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.