Question

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

Answer 1

Probability that their mean salary is greater than $32,500 is 0.0475.

Step-by-step explanation:

We are given 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.

Also, 100 teachers are randomly selected.

Let $\bar X$ = sample mean salary

Now, the z score probability distribution for sample mean is given by;

         Z = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\mu$ = mean salary of elementary school teachers in the United States = $32,000

           $\sigma$ = standard deviation = $3,000

           n = sample of teachers = 100

So, probability that their mean salary is greater than $32,500 is given by = P($\bar X$ > $32,500)

P($\bar X$ > $32,500) = P( $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ > $\frac{32,500-32,000}{\frac{3,000}{\sqrt{100} } }$ ) = P(Z > 1.67) = 1 - P(Z $\leq$ 1.67)

                                                                  = 1 - 0.9525 = 0.0475

Hence, probability that their mean salary is greater than $32,500 is 0.0475.

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