Question

(1 point) 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 27 dollars and a standard deviation of 8 dollars.

A. What proportion of the bank's Visa cardholders pay more than 29 dollars in interest? Proportion =

B. What proportion of the bank's Visa cardholders pay more than 36 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 18% of the bank's Visa cardholders? Interest Payment =

Answer 1

Part a)

X ~ N ( µ = 27 , σ = 8 )
P ( X > 29 ) = 1 - P ( X < 29 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 29 - 27 ) / 8
Z = 0.25
P ( ( X - µ ) / σ ) > ( 29 - 27 ) / 8 )
P ( Z > 0.25 )
P ( X > 29 ) = 1 - P ( Z < 0.25 )
P ( X > 29 ) = 1 - 0.5987
P ( X > 29 ) = 0.4013

Part b)

X ~ N ( µ = 27 , σ = 8 )
P ( X > 36 ) = 1 - P ( X < 36 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 36 - 27 ) / 8
Z = 1.125
P ( ( X - µ ) / σ ) > ( 36 - 27 ) / 8 )
P ( Z > 1.125 )
P ( X > 36 ) = 1 - P ( Z < 1.125 )
P ( X > 36 ) = 1 - 0.8697
P ( X > 36 ) = 0.1303

Part c)

X ~ N ( µ = 27 , σ = 8 )
P ( X < 18 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 18 - 27 ) / 8
Z = -1.125
P ( ( X - µ ) / σ ) < ( 18 - 27 ) / 8 )
P ( X < 18 ) = P ( Z < -1.125 )
P ( X < 18 ) = 0.1303

part d)

X ~ N ( µ = 27 , σ = 8 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.18 = 0.82
To find the value of x
Looking for the probability 0.82 in standard normal table to calculate Z score = 0.9154
Z = ( X - µ ) / σ
0.9154 = ( X - 27 ) / 8
X = 34.3232
P ( X > 34.3232 ) = 0.18