Loans that are 60 days or more past due are considered serio

Loans that are 60 days or more past due are considered seriously delinquent. The Mortgage Bankers Association reported that the rate of seriously delinquent loans has an average of 9.1% (The Wall Street Journal, August 26, 2010). Let the rate of seriously delinquent loans follow a normal distribution with a standard deviation of 0.80%.

a.

What is the probability that the proportion of seriously delinquent loans is above 8%?

b.

What is the probability that the proportion of seriously delinquent loans is between 9.5% and 10.5%?

Solution

a)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    8      
u = mean =    9.1      
          
s = standard deviation =    0.8      
          
Thus,          
          
z = (x - u) / s =    -1.375      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -1.375   ) =    0.915434278 [answer]

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b)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    9.5      
x2 = upper bound =    10.5      
u = mean =    9.1      
          
s = standard deviation =    0.8      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    0.5      
z2 = upper z score = (x2 - u) / s =    1.75      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.691462461      
P(z < z2) =    0.959940843      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.268478382 [ANSWER]

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