Question

1. The length of incoming calls at the call center of a major telecommunication service provider follows a Normal distribution with an average of 7.5 minutes. If the standard deviation of the distribution is 5 minutes, answer the following questions: (8 points)
   
   1. What is the probability that the length of an incoming call is longer than 9.5 minutes?
   2. What is the probability that the length of an incoming call is shorter than 6 minutes?
   3. What is the probability that the length of an incoming call is between 6.5 to 25 minutes?
   4. 25% of the incoming calls is longer than how many minutes?
      
2. The servicing time at the drive-through lane of a fast food restaurant follows an exponential distribution. The average servicing time is 1.5 minutes. (4 points)
   
   1. What is the probability that it takes more than 2 minutes to service a customer at the drive-through lane?
   2. What percent of customers at the drive-through lane will take between 1 to 2 minutes to service?

Answer 1

1)a) P(X > 9.5)

  

= P(Z > 0.4)

= 1 - P(Z < 0.4)

= 1 - 0.6554

= 0.3446

b) P(X < 6)

  

= P(Z < -1)

= 0.1587

c) P(6.5 < X < 25)

= P(0.2 < Z < 3.5)

= P(Z < 3.5) - P(Z < 0.2)

= 0.9998 - 0.5793

= 0.4205

d) P(X > x) = 0.25

  

or, x = 0.67 * 5 + 7.5

or, x = 10.85

a) = 1.5

= 1/1.5

P(X > 2) = e^-x = e^(-1/1.5 * 2) = e^(-2/1.5) = 0.2636

b) P(1 < X < 2)

= P(X < 2) - P(X < 1)

= (1 - e^(-1/1.5 * 2)) - (1 - e^(-1/1.5 * 1))

= 0.7364 - 0.4866

= 0.2498