Question

The mean annual income for people in a certain city (in thousands of dollars) is 41, with standard deviation of 29. It is normally distributed.

If your income is in the middle 60% of incomes in the city, you are put into special tax bracket. Between what two incomes must a single person make to be in this tax bracket?

Answer 1

If $X\sim N(\mu,\sigma^2),$ then $Z=\dfrac{X-\mu}{\sigma}\sim N(0, 1)$

$$\mu = 41, \sigma= 29.$$

$$P(\mu-\Delta x<X<\mu+\Delta x)=P(z^*-\Delta z<Z<z^*+\Delta z)=0.6$$

$$P(-0.841621<Z<0.841621)=0.6$$

$$\dfrac{X_1-\mu}{\sigma}=-0.841621$$

$$X_1=41+29(-0.841621)=16.59$$

$$\dfrac{X_2-\mu}{\sigma}=0.841621$$

$$X_2=41+29(0.841621)=65.41$$

The mean annual income for people in a certain city (in thousands of dollars) is between 16.59 and 65.41.