a mean of $70,000 and standard deviaiton of $20,000?

Question

The amounts of money requested on home loan applications at Down River Federal Savings
follow the normal distribution, with a mean of $70,000 and a standard deviation of
$20,000. A loan application is received this morning. What is the probability:
a. The amount requested is $80,000 or more?
b. The amount requested is between $65,000 and $80,000?
c. The amount requested is $65,000 or more?

Answer 1

For each of these, you want to convert to the Z-values (normal disrtibution with mean 0 and standard deviation 1) which are probably in the back of your book.

a) P(L>80,000)
= P((L-70000)/20000 > (80000 - 70000)/20000)
= P(Z > .5) = .3085

(In case you want more steps, P(Z > .5) = P(Z > 0) - P(0 < Z < .5) = .5 - .1915 = .3085. The .1915 is the number for 0.5 in the Z tables)

b) P(65000 < L < 80000)
= P((65000-70000)/20000 < Z < (80000 - 70000)/20000)
= P(-.25 < Z < .5) = P(-.25 < Z < 0) + P(0 < Z < .5)
= P(0 < Z < .25) + P(0 < Z < .5) = .0987 + .1915 = .2902

c) P(65000 < L) = P(-.25 < Z)
= P(-.25 < Z < 0) + P(Z > 0) = .0987 + .5 = .5987

PS The first respondent gave the right answer for $90000 or more, but the requested $80000 is only half a standard deviation above the mean.

Answer 2

Dude,
I have decided to reteach myself statistics, because there seems to be a need around here. I hope that i can help.

It always helps to draw a graph, to help you to visualize.

Draw a bell curve, then draw a line down the middle. This is the mean. The chart with the z (number of standard deviations) only gives the area under the curve form the center line to however many "z"s or standard deviations to one side of the center or mean. so

a.) Label your mean as $70,000, then draw a line a little to the right and label that $80,000. The distance between the mean and the $80,000 is $10,000. Since our standard deviation is $20,000, this distance is 10,000/20,000 or 1/2 of one standard deviation (z = 1/2). Now, you can look on the z-table for z=0.5 and you find that the area under the curve from the mean to $80,000. the table says that it is .1913 or 19.13%. But this is not your answer. You were asked for the area ABOVE $80,000. Well, you know that everthing above the mean is 50 %, so the area from 80 K upwards must be 50% minus the area between the mean and 80 K, or 50% - 19.13%

answer is 30.87%

b.) this method and table only do things from the center out, so in this question, you must break the area you want to find in half, split at the mean.

Draw a line a little to the left of the mean and label it $65,000. To find the area form 65 K to 80 K we really need to find the area between 65 K and the mean. We will add this to the area from the mean to 80 K, which we already know is 19.13%

z for 65 K to the mean is (70,000-65,000)/20,000 or 0.25. From the chart at z=0.25, area equals 0.0987 or 9.87%

add this to 19.13% (area from mean to 80 k ) and you get

answer = 29 %

c) Now they want the area under the curve from 65 K upward. You already know the area fro 65 K to the mean (9.87% from above) and you know that from the mean upwards is 50% of the area, just add them. 50% + 9.87% =
answer = 59.87%
I hope this helps, dude

Answer 3

a. (80 - 70)/20 = 0.5σ P(L>$80,000) = 0.3085
b. (80 - 65)/20 = 0.75σ
P($65,000 < L < $80,000) = 0.2902
c. P(L > $65,000) = 0.5987

Answer 4

Downriver Federal

Answer 5

Great point, I'm interested to know more too

Answer 6

Thank you everyone for answering.