Question

problem 7
full - time ph.d. students receive an average of $12,500 per year. if the salaries are normally distributed with a standard deviation of $980, find the probability that a student makes more than $10,000

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 10000\), \(\mu=12500\), and \(\sigma = 980\).

$$ z=\frac{10000 - 12500}{980}=\frac{- 2500}{980}\approx - 2.55 $$

Step2: Find the probability

We want to find \(P(X>10000)\). Since \(P(X>x)=1 - P(X\leq x)\), and for a normal distribution \(P(X\leq x)=\Phi(z)\) (the cumulative distribution function of the standard normal distribution).
From the standard normal table, \(\Phi(-2.55)=0.0054\)
So \(P(X > 10000)=1-\Phi(-2.55)\)

$$ P(X>10000)=1 - 0.0054=0.9946 $$

Answer:

The probability that a student makes more than \(\$10000\) is \(0.9946\)