Question

select the correct answer. the income distribution in a county is a normal distribution with a mean income of $10,000. the top 2.5% of the wage earners earn $18,000 or more. which sentence most closely summarizes this information? a. 95% of the wage earners earn less than $14,000 each. b. 68% of the wage earners earn less than $14,000 each. c. 16% of the wage earners earn less than $14,000 each. d. 84% of the wage earners earn less than $14,000 each.

Explanation:

Step1: Recall properties of normal distribution

In a normal distribution, about 95% of the data lies within 2 standard - deviations of the mean, and about 68% lies within 1 standard - deviation of the mean. The top 2.5% of the data lies 2 or more standard - deviations above the mean. If the mean is $\mu = 10000$ and the value corresponding to the top 2.5% is $x = 18000$, we can find the standard deviation $\sigma$ using the z - score formula $z=\frac{x-\mu}{\sigma}$. The z - score corresponding to the top 2.5% (or the 97.5th percentile) is $z = 2$. So, $2=\frac{18000 - 10000}{\sigma}$, which gives $\sigma=\frac{18000 - 10000}{2}=4000$.

Step2: Find the z - score for $x = 14000$

Using the z - score formula $z=\frac{x-\mu}{\sigma}$, with $x = 14000$, $\mu = 10000$ and $\sigma = 4000$, we get $z=\frac{14000 - 10000}{4000}=1$.

Step3: Determine the percentage of data

The area to the left of $z = 1$ in a standard normal distribution is approximately 84%. This means that 84% of the wage earners earn less than $14000$ each.

Answer:

D. 84% of the wage earners earn less than $14,000$ each.