Question

How do I solve a normal distribution problem?

Answer 1

When you have a normal distribution with mean µ and standard deviation σ, any x value in the distribution can be converted to a z value by

z = (x - µ)/σ

The z value is then used in a standard normal distribution with mean 0 and standard deviation 1. Tables in textbooks are based on the standard normal distribution.

Cumulative tables usually start with z = 0, which corresponds to the mean. Half of the area under the curve is to the left of the mean, so the table value for z = 0 is 0.5.

Your problem is asking for the middle 80% of the area under the curve, so you need 40% of the area to the right of the mean and 40% of the area to the left of the mean.

Since 50% of the area is to the left of the mean, and 40% is to the right of the mean, you need a z score that gives a total of 90% of the area under the curve. Since the distribution is symmetric, the area you're after will be between -z and +z.

From the table, you'll find that 90% of the area lies to the left of z = 1.282. So we need the x's that correspond to z = -1.282 and +1.282.

From the equation above, we see that x = zσ + µ, so

x(lower) = (-1.282)(15,000) + 246,300

= -19,230 + 246,300 = $227,070

x(upper) = 1.282(15,000) + 246,300

= 19,230 + 246,300 = $265,530