select the correct answer.\nthe income distribution in a county is a normal distribution with a mean income…

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 99.7% lies within 3 standard - deviations of the mean. The top 2.5% of the data lies above 2 standard - deviations from the mean. Given mean $\mu = 10000$ and a value $x = 18000$ corresponding to the top 2.5% of the data. We first 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: Calculate z - score for $x = 14000$

The z - score formula is $z=\frac{x-\mu}{\sigma}$. Substitute $\mu = 10000$, $\sigma = 4000$, and $x = 14000$ into the formula: $z=\frac{14000 - 10000}{4000}=\frac{4000}{4000}=1$.

Step3: Find the percentage of data below $z = 1$

Looking up the z - score of 1 in the standard normal distribution table, the area to the left of $z = 1$ (the percentage of data less than the value corresponding to $z = 1$) is approximately 0.8413 or 84%.

Answer:

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