Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $32,000 and a standard deviation of $3000. If 100 teachers are randomly selected, find the probability
That their mean salary is greater than $32,500.
A)0.0475
B)0.9525
C)0.3312
D)0.1312

Question

Quizplus · Accepted Answer

To find the probability that the mean salary of 100 randomly selected teachers is greater than $32,500, we use the formula for the standard error of the mean: $SE = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the standard deviation and $n$ is the sample size. Here, $\sigma = $3000$ and $n = 100$, so $SE = \frac{3000}{\sqrt{100}} = \frac{3000}{10} = $300$.Next, we calculate the z-score, which is $\frac{\bar{x} - \mu}{SE}$, where $\bar{x}$ is the sample mean and $\mu$ is the population mean. Substituting the given values, we get $\frac{32500 - 32000}{300} = \frac{500}{300} = 1.67$.Looking up the z-score of 1.67 in the standard normal distribution table or using a calculator, we find that the area to the left of this z-score is approximately 0.9525. Since we're interested in the probability that the mean salary is greater than $32,500, we subtract this value from 1 to get $1 - 0.9525 = 0.0475$. This represents the probability that the mean salary of 100 randomly selected teachers is greater than $32,500.

Assume That the Salaries of Elementary School Teachers in the United