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 29 dollars and a standard deviation of 9 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 41 dollars in interest?

Proportion =

C. What proportion of the bank's Visa cardholders pay less than 19 dollars in interest?

Proportion =

D. What interest payment is exceeded by only 18% of the bank's Visa cardholders?

Interest Payment =

Answer 1

Answer:

a) Proportion = 41.3%

b) Proportion = 9.18%

c) Proportion = 13.35%

d) Interest payment = $37.28

Explanation:

We have normal distribution with mean = 29 and standard deviation = 9

a) The proportion of the bank’s Visa cardholders pay more than $31 in interest is:

P(X > 31) = $(\frac{X-29}{9}>\frac{31 -29}{9})$ = P (Z > 0.22) = 1 - P (Z ≤ 0.22) =  

1 - 0.58706 = 0.41294 = 41.3%

The proportion of the bank's Visa cardholders pay more than 31 dollars in interest is 41.3%.

b) The proportion of the bank’s Visa cardholders pay more than $31 in interest is:

P(X > 41) = $(\frac{X-29}{9}>\frac{41 -29}{9})$ = P (Z > 1.33) = 1 - P (Z ≤ 1.33) =  

1 - 0.90824 = 0.09176 = 9.176% ≈ 9.18%

The proportion of the bank's Visa cardholders pay more than 31 dollars in interest is 9.18%.

c) The proportion of the bank’s Visa cardholders pay more than $31 in interest is:

P(X > 19) = $(\frac{X-29}{9}>\frac{19 -29}{9})$ = P (Z < -1.11) = 1 - P(Z ≤ -1.11)) =0.13350 = 13.35%

The proportion of the bank's Visa cardholders that paid less than 19 dollars in interest is 13.35%.

d) Let's suppose this amount of payment is Y:

Therefore P(X > Y) = 0.18

so P(X < Y) = 0.82

Utilizing standard normal approximation

P(X ≤ Y) = $(\frac{X-29}{9}\leq \frac{Y -29}{9})$ = P (Z ≤ $\frac{Y-29}{9}$)  =  0.82

Form the standard normal table we find that $\frac{Y-29}{9}$ = 0.92

Therefore,

Y - 29 = 9×0.92

Y - 29 = 8.28

Y = 8.28 + 29 = 37.28

Therefore $37.28 of interest payment is exceeded by only 18% of the bank's Visa cardholders.