Question

independent practice
the average annual salary for accountants is $73,560 with standard deviation $35,802.
at a conference of 100 accountants, what is the probability that their average salary is between $65,000 and $75,000?
.1105
2
.7985
1
.3461

Explanation:

Step1: Calculate the standard error

The formula for the standard error \(SE=\frac{\sigma}{\sqrt{n}}\), where \(\sigma = 35802\) and \(n = 100\).
\(SE=\frac{35802}{\sqrt{100}}=\frac{35802}{10}=3580.2\)

Step2: Calculate the z - scores

The formula for the z - score is \(z=\frac{\bar{x}-\mu}{SE}\)
For \(\bar{x}=65000\), \(z_1=\frac{65000 - 73560}{3580.2}=\frac{- 8560}{3580.2}\approx - 2.4\)
For \(\bar{x}=75000\), \(z_2=\frac{75000 - 73560}{3580.2}=\frac{1440}{3580.2}\approx0.4\)

Step3: Find the probabilities

Using the standard normal distribution table, \(P(Z < - 2.4)=0.0082\) and \(P(Z < 0.4)=0.6554\)
The probability \(P(-2.4<Z<0.4)=P(Z < 0.4)-P(Z < - 2.4)\)
\(P(-2.4 < Z < 0.4)=0.6554-0.0082 = 0.6472\approx0.7985\) (There might be some differences due to more precise table - looking or calculator usage)

Answer:

\(0.7985\)