Calls to a customer service center last on average 26 minute

Calls to a customer service center last on average 2.6 minutes with a standard deviation of 1.2 minutes. An operator in the call center is required to answer 89 calls each day. Assume the call times are independent.

a. What is the expected total amount of time in minutes the operator will spend on the calls each day? Give an exact answer.  

2.6*89=231.4

b. What is the standard deviation of the total amount of time in minutes the operator will spend on the calls each day? Give your answer to four decimal places.

1.2 *sqrt(89)=11.3208

c. What is the approximate probability that the total time spent on the calls will be less than 226 minutes? Give your answer to four decimal places. Use the standard deviation as you entered it above to answer this question.  

? (can you also link me to table you are using)

d. What is the value c such that the approximate probability that the total time spent on the calls each day is less than c is 0.97? Give your answer to four decimal places. Use the standard deviation as you entered it above to answer this question.

?

Solution

a)

Mean = 2.6*89 = 231.4

b)

Standard deviation = s*Sqrt(n) = 1.2*sqrt(89) = 11.32077736 [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    226      
u = mean =    231.4      
          
s = standard deviation =    11.32077736      
          
Thus,          
          
z = (x - u) / s =    -0.476999046      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -0.476999046   ) =    0.316681401 [ANSWER]

[You can also use the NORM.DIST function in Excel to get left tailed areas.]

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d)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.97      
          
Then, using table or technology,          
          
z =    1.880793608      
          
As x = u + z * s,          
          
where          
          
u = mean =    231.4      
z = the critical z score =    1.880793608      
s = standard deviation =    11.320777      
          
Then          
          
x = critical value =    252.692045   [ANSWER]  

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You can use this normal distribution table, if you want to use a table:

https://www.stat.tamu.edu/~lzhou/stat302/standardnormaltable.pdf