Question

Please show all work. Thank you!

The amount of money requested on home loan applications at America Bank follows normal distribution, with a mean of $180,000 and a standard deviation of $6,000. A loan application is received this morning.

a) What is the probability the amount requested is $190,000 or more?

b) What is the probability the amount requested is between $178,000 and $190,000?

c) What is the probability the amount requested in below $178,000?

d) How much is requested on the largest 5% of the loans?

e) How much is requested on the smallest 5% of the loans?

Answer 1

Solution:

We are given that home loan applications at America Bank follow normal distribution, with a mean of $180,000 and a standard deviation of $6,000.

Mean = 180000

SD = 6000

Part a

Here, we have to find P(X>190000)

P(X>190000) = 1 – P(X<190000)

Z = (X – mean) / SD

Z = (190000 - 180000)/6000

Z = 1.666667

P(Z<1.666667) = P(X<190000) = 0.95221

(by using z-table)

P(X>190000) = 1 – P(X<190000)

P(X>190000) = 1 – 0.95221

P(X>190000) = 0.04779

Required probability = 0.04779

Part b

Here, we have to find P(178000<X<190000)

P(178000<X<190000) = P(X<190000) – P(X<178000)

First find P(X<190000)

Z = (X – mean) / SD

Z = (190000 - 180000)/6000

Z =1.666667

P(Z<1.666667) = P(X<190000) = 0.95221

(by using z-table)

Now find P(X<178000)

Z = (X – mean) / SD

Z = (178000 - 180000)/6000

Z = -0.33333

P(Z<-0.33333) = P(X<178000) = 0.369441

(by using z-table)

P(178000<X<190000) = P(X<190000) – P(X<178000)

P(178000<X<190000) = 0.95221 - 0.369441

P(178000<X<190000) = 0.582769

Required probability = 0.582769

Part c

Here, we have to find P(X<178000)

Z = (X – mean) / SD

Z = (178000 - 180000)/6000

Z = -0.33333

P(Z<-0.33333) = P(X<178000) = 0.369441

(by using z-table)

Required probability = 0.369441

Part d

Here, we have to find the amount for the largest 5% of the loans.

The z critical value for largest 5% is 1.644854 (by using z-table).

X = Mean + Z*SD

X = 180000 + 1.644854*6000

X = 189869.1

Required answer: $189,869.1

Part e

Here, we have to find the amount for the smallest 5% of the loans.

The z critical value for smallest 5% is -1.64485 (by using z-table).

X = Mean + Z*SD

X = 180000 - 1.644854*6000

X = 170130.9

Required Answer: $170,130.9