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 $66,000 and a standard deviation of $20,000. (Round z-score computation to 2 decimal places and the final answers to the nearest whole dollars.) a.

aWhat is the minimum amount requested on the largest 8% of loans? $

b. What is the maximum amount requested on the smallest 11% of loans?

Answer 1

Ans a. let us suppose amount of money requested on home loan is represented by a random varianble X and we are given that X~N($66000,$20000^2).

therefore Z=x-mean/standard deviation ~N(0,1).

Minimum amount requested on the largest 8% of loans implies that we need to find x for which the P(X>=x)=0.08 , which can be written as:

P(Z>=x-mean/sd)=0.08 =1-P(Z<x-mean/sd)

P(Z<x-mean/sd) =0.92

Now using table of standard normal probabilities we get x-mean/sd=1.41(approx). Now putting the value of mean and sd we get the value of x as $94200.

Therefore the minimum amount of loan requested on the largest 8% of the loans is $94200.

Ans b. This time we need to find the maximum amount requested on the smallest 11% of loans, this  implies that we need to find x for which  

P(X<=x)=0.11 = P(Z<=x-mean/sd)=0.11

  Using the table of standard normal probabilities we get the value of x-mean /sd = -1.227= -1.23 (approx).

putting the value of mean and sd in the above equation we get the value of x as $41400.

Therefore the maximum amount of loan requested on the smallest 11 % of the loans is $41400.