An executive at Bank A is concerned that the bank is losing

An executive at Bank A is concerned that the bank is losing customers because mortgage applications are taking too long to be approved. The national average, according to an industry newsletter, is six business days. If the executive decides that this bank is taking longer than the national average then she will make major changes to the mortgage department, including demoting or firing the department head. She instructs a vicepresident to investigate the situation. The VP collects a random sample of 12 recent mortgage applications from 9 of the bank’s many branches. The time to process each application (number of business days) is recorded. The 108 observations are listed below.

Bank A 8, 6, 7, 7, 3, 12, 11, 11, 7, 7, 8, 3, 5, 5, 6, 9, 6, 4, 3, 4, 4, 9, 7, 8, 8, 7, 6, 9, 8, 7, 7, 6, 4, 5, 5, 2, 7, 9, 11, 8, 5, 7, 9, 8, 9, 4, 7, 2, 11, 9, 6, 8, 2, 5, 5, 9, 8, 7, 4, 2, 6, 9, 8, 9, 7, 6, 10, 5, 10, 7, 9, 6, 8, 10, 5, 9, 5, 5, 9, 5, 4, 11, 10, 5, 7, 9, 11, 5, 5, 10, 7, 5, 11, 10, 12, 8, 8, 4, 6, 8, 6, 7, 7, 12, 5, 6, 8, 7.

1) Construct a 99% confidence interval for the mean mortgage processing time. Express the confidence interval in the form: “The average processing time for mortgages is ?? days. This survey is accurate to within ±?? days, ?? times out of ??”.

2) The executive decides that she wants to have a 99% confidence interval with a margin of error of no more than 0.25 days. How many mortgage applications should she sample?

3) Based on the results of the significance test, what decision should the bank executive make? Make sure to state this in a way that can be understood by someone who doesn’t know any statistics.

Solution

1.

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    7.018518519          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    2.391501626          
n = sample size =    108          
              
Thus,              
              
Lower bound =    6.425762623          
Upper bound =    7.611274414          
              
Thus, the confidence interval is              
              
(   6.425762623   ,   7.611274414   ) [ANSWER]

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b)

Note that      
      
n = z(alpha/2)^2 s^2 / E^2      
      
where      
      
alpha/2 = (1 - confidence level)/2 =    0.005  
      
Using a table/technology,      
      
z(alpha/2) =    2.575829304  
      
Also,      
      
s = sample standard deviation =    2.391501626  
E = margin of error =    0.25  
      
Thus,      
      
n =    607.1493058  
      
Rounding up,      
      
n =    608   [ANSWER]

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C)

As the whole confidence interval is greater than 6, then we conclude:

There is significant evidence to say that the average of this bank is taking longer than the average, and it didn\'t just happen by chance.