Question

The amounts of money requested on home loan applications at Down River Federal Savings follow the normal distribution, where the amount requested for home loans followed the normal distribution with a mean of $80,000 and a standard deviation of $24,000. (Round z-score computation to 2 decimal places and the final answers to the nearest whole dollars.)

a. What is the minimum amount requested on the largest 2% of loans?

$

b. What is the maximum amount requested on the smallest 10% of loans?

Answer 1

Amount of money requested, X ~ N(80000, 24000²)

Ans a)  Largest 2% of loans lie at and beyond the 98 percentile of the Normal curve, i.e. those which have a probability >= 0.98

Now, the Z-score corresponding to 0.98 = - ( Z-score corresponding to 0.02) (using the symmetricity of standard normal curve about origin)

From the Z-score table, P(Z < -2.879) = 0.02 => P(Z > 2.879) is also 0.02

Now, the minimum amount of the largest 2% of loans corresponds to the Z-score of 2.879

Hence, if x is the loan amount in the normal distribution corresponding to this Z-score, we have

( x - 80000) / 24000 = 2.879 => x = 1,49,096$

Ans b)  Similar to above, Z-score of smallest 10% of loans corresponds to the 10th percentile (or probability of 0.1) on the Normal curve.

Z-score corresponding to 0.1 = -1.282

So, if x is the maximum amount on the smallest 10% loans, x has a Z-score of -1.282

(x - 80000) / 24000 = -1.282 => x = 49.232$