How do I solve for proportion and Interest?

Question

Because of the relatively high interest rates, most consumers attempt to pay off their credit card bills promptly. However, this is not always possible. An analysis of the amount of interest paid monthly by a bank's Visa cardholders reveals that the amount is normally distributed with a mean of 28 dollars and a standard deviation of 7 dollars.

A. What proportion of the bank's Visa cardholders pay more than 31 dollars in interest?

Proportion = 

B. What proportion of the bank's Visa cardholders pay more than 39 dollars in interest?

Proportion = 

C. What proportion of the bank's Visa cardholders pay less than 18 dollars in interest?

Proportion = 

D. What interest payment is exceeded by only 19% of the bank's Visa cardholders?

Interest Payment = 

Answer 1

These can all be obtained using the area under the normal distribution (bell-shaped) curve.

Let z be the number of standard deviations (sigma) x is different from the mean

z = (x - mean)/sigma

Use a table of normal distribution values (better yet, cumulative distribution, the area under the curve):

P(z) = percentage smaller than or equal to z.

For a normal distribution ("Gaussian"), the fraction less than or equal to the mean, thus z=0,

is also the median, 50%, 0.50.

P(z=1) is a standard deviation above the mean, P(1)=0.84.

a) x=31, z=(31-28)/7=0.43. P(0.43) = 0.666 = 66.6% of the users pay less than 31

so 33.4% pay more.

b) x=39, z=(39-28)/7=1.57. P(1.57) = 0.941 = 94.1%

so 1-0.941= 0.059 = 5.9% of the users pay more than 39.

c) x=18, z=(18-28)/7=-1.43. P(-1.43) = 0.076 = 7.6% of the users pay less than 18.

d) Use z = (x - mean)/sigma, but get z from the distribution table for P(z),

P(z) = 81% is 0.86 sigma above the median, so we calculate the

x = mean - 0.90 sigma = 28+0.0.86*7 = 34.0.

Thus, 19% have interest costs of 34.0 or more.