How to find bottom 10% of income of 33,000

Question

If income is 33000. with a standard deviation of $9000 using a normal distribution with specified mean and standard deviation how to calulate the income cut off the bottom 10%

Answer 1

We need to calculate the Z score corresponding to the lower 10% of the distribution. The Z score will be negative and it will tell us how many SD's below the mean we need to go. The magnitude of this Z score (the positive equivalent of the negative value) would give us the top 10% because the Normal Distribution is symmetrical. Inspect the body of the ND table for the value 0.9 (or as close as possible). This will give us the Z score for 90% of incomes, so that would leave 10% in the top income bracket. It turns out that Z=1.28 approx corresponds to 0.9. So -1.28 corresponds to the 10% in the bottom income bracket. Now we use the formula (X-33000)/9000=-1.28 when the mean is $33000 and SD=$9000. X is the income cutoff.

X-33000=-11520, X=33000-11520=$21,480. This is the cutoff value of the income (approximately).