Assume that the random variable X is normally distributed with mean 70 and standard deviation 8. Find the 40th percentile for X.

Answer 1

This can be done with a normal table or technology like a TI-84 calculator. On the calculator, there is a probability distribution menu that is accessed by 2ND VARS. You'll see InvNorm( as the third option down. There are 3 fields to fill in. The first is the desired left-tail area. In this case, since we want the 40th percentile, it is 0.4. The next is mean, which is 70, and last is standard deviation, which is 8. The answer the calculator gives is about 67.973.

To do this using a normal table, we need to do the reverse of the typical procedure. We need to begin with the left-tail area by looking on the table for the area closest to 0.4, and its corresponding z-score. This is -0.25, which has a left-tail area of 0.40129. Then we use the z-score formula to solve for the missing x value.

^{(x - μ)} / _σ = z

^{(x - 70)} / ₈ = -0.25

x - 70 = -2

x = 68

Answer 2

Prob ( z < k) = 0.4

z=-0.25335 by reverse table lookup

z = (x - mean)/stddev

-0.25335 = (x - 70)/ 8

-0.25335 * 8+ 70= x

x = 67.9732

Answer 3

Find z value from normal distribution table that corresponds to an area of .4 to the left (-.253)

Now solve for x given that z = (x-μ)/σ or x = μ + σz where μ is mean and σ is st. dev.

Plug in z, mean, and standard dev.

Hope that helps.