**Question 1**: The owners of Spiffy Lube want to offer their customers a 10-minute guarantee on their standard oil change service. If the oil change takes longer than 10 minutes to complete, the customer is given a coupon for a free oil change at the next visit. Based on past history, the owners believe that the time required to complete an oil change has a normal distribution with a mean of 8.6 minutes and a standard deviation of 1.2 minutes.

a. [5 points] What percentage of customers will receive a free oil change coupon?

b. [5 points] If management wants to limit the percentage of customers receiving a coupon to no more than 1 out of every 25 customers on average, what should they change the guarantee time to?

a. [10 points] Suppose management could improve the process by reducing the mean time required for an oil change (but keeping the standard deviation the same). How much change in the mean service time would be required to allow for a 10-minute guarantee that gives a coupon to no more than 1 out of every 25 customers on average?

**Question 2**: Suppose it has been observed that touchdowns scored in a game by your favorite football team are Poisson random variables. The rate is = 4 per game. We know from the lecture that the time until the next touchdown follows an exponential distribution. Assume a football game is 60 minutes long (that is we are concerned only with the regular playing time).

a. [5 points] What is the expected time in minutes until the next touchdown is scored by your favorite team?

b. [5 points] What is the probability that the next touchdown will be scored in the next 10 minutes by your favorite team?

c. [5 points] What is the probability that there will be no more touchdowns scored by your favorite team given that we are in the 50th minute of the game?

**Question 3**: [20 points] You are evaluating two portfolios for your personal investment on the basis of their return. Portfolio A's yearly return is normally distributed with a mean of 0.05 and a standard deviation of 0.07. Portfolio B's yearly return is normally distributed with a mean of 0.04 and a standard deviation of 0.06. Based on this information, what is the probability that over the next year, the return on portfolio A is lower than the return on portfolio B?

Subject | Mathematics |

Due By (Pacific Time) | 06/21/2014 900pm |

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