Question

problem 8
the average salary of college professor in new york is $83,000. the standard deviation is $7,000. what is the probability that a professor makes between $75,000 and 87,000?

Explanation:

Step1: Calculate z - scores

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(\mu\) is the mean, \(\sigma\) is the standard deviation, and \(x\) is the value.
For \(x = 75000\), \(z_1=\frac{75000 - 83000}{7000}=\frac{- 8000}{7000}\approx - 1.14\)
For \(x = 87000\), \(z_2=\frac{87000 - 83000}{7000}=\frac{4000}{7000}\approx0.57\)

Step2: Use the standard normal distribution table

We know that \(P(-1.14<Z<0.57)=P(Z < 0.57)-P(Z<-1.14)\)
From the standard - normal table, \(P(Z < 0.57)=0.7157\), \(P(Z<-1.14) = 0.1271\)

Step3: Calculate the probability

\(P(-1.14 < Z < 0.57)=0.7157-0.1271 = 0.5886\)

Answer:

\(0.5886\)