Question

analyzing income percentages
tax data
company tax data for employees
a companys 2022 data for employee incomes shows a mean annual income of $50,000, and the standard deviation of annual income is $8,000. assume the incomes are normally distributed.
annual income tax data distribution (2022)
what percentage of the companys employees have an annual income between $42,000 and $58,000?
95.4%
34%
68%
99.7%

Explanation:

Step1: Calculate z - scores

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $\mu = 50000$ is the mean and $\sigma = 8000$ is the standard deviation.
For $x = 42000$, $z_1=\frac{42000 - 50000}{8000}=\frac{- 8000}{8000}=-1$.
For $x = 58000$, $z_2=\frac{58000 - 50000}{8000}=\frac{8000}{8000}=1$.

Step2: Use the empirical rule for normal distribution

The empirical rule states that for a normal distribution, approximately 68% of the data lies within 1 standard - deviation of the mean, that is, between $z=-1$ and $z = 1$.

Answer:

68%