Question

1. Suppose that the time it takes you to drive to work is a normally distributed random variable with a mean of 20 minutes and a standard deviation of 4 minutes.

a. the probability that a randomly selected trip to work will take more than 30 minutes equals: (5 pts)

b. the expected value of the time it takes you to get to work is: (4 pts)

c. If you start work at 8am, what time should you leave your house if you want no more than a 2.5% chance that you will be late for work? (6 pts)

2. The temperature in a southern Arizona city has a uniform distribution with a range from 82 degrees to 110 degrees.

a. Draw a picture appropriate to this probability density function (pdf). (5 pts)

b. Write an equation for this pdf. (4 pts)

c. What is the median temperature in this city? (4 pts)

d. What is the 90^th percentile temperature? (4 pts)

e. What is the probability that the temperature on a randomly selected day exceeds 100? (5 pts)

f. Suppose that on a given day, the temperature reaches 90 by early in the morning. What is the probability that the temperature that day will exceed

98 degrees? (5 pts)

3. The time needed to find a parking space at a college campus is approximately normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes.

a. What is the probability that a person can find a parking space in less than 9 minutes? (5 pts)

b. If you arrive on campus 14 minutes and 30 seconds before class, what is the probability that you will be late for class because you cannot find a parking place? (5 pts)

c. Dr. Stern threatens to lower your class grade by one full letter grade if you arrive late one more time. You assess the situation and are willing to run the risk of that happening 5% of the time. How much time should you leave to park? (6 pts)

4. Twelve percent of all army recruits are too short for parade duties at national cemeteries. The minimum height for this duty is 70 inches. If the distribution of recruits’ heights is normal with a standard deviation of 3 inches, what is the average recruit height? (6 pts)

5. Tuna cans filled by a certain machine should weigh, on average, 6.5 ounces with a standard deviation of 0.01 ounces. This is true if the machine is working properly. Past evidence demonstrates that the distribution of weights is normal.

a. Quality inspectors will accept all cans between 6.475 and 6.525 ounces. What proportion of the cans will pass this quality inspection? (6 pts)

b. Suppose the packing machine is mis-calibrated and fills cans with an average of 6.495 ounces. What is the probability that a can will pass inspection, based upon the inspector’s rule from part (a), even though the machine is not filling according to the label specifications? (6 pts)

6. A company has a lump-sum incentive plan for its sales staff that is dependent on their level of sales. If they sell less than $100,000 per year they receive a $1000 bonus; from $100,000 to $200,000 they receive $5,000; and above $200,000 they receive $10,000. If the annual sales (per salesperson) follow a normal distribution with mu=$160,000 and sigma=$40,000:

a. Find the proportion of salespeople who receive a $1000 bonus.   (4 pts)

b. Find the proportion of salespeople who receive a $5000 bonus.   (4 pts)

c. Find the proportion of salespeople who receive a $10000 bonus.   (4 pts)

d. What is the mean value of the bonus payout for the company?   (6 pts)

7. You take your laundry to a laundromat and pay a quarter each time you run the dryer. After much study you have determined that the length of time the dryer runs on one quarter is a continuous uniform random variable bounded by 6 and 12 minutes. A pile of wet laundry needs 19 minutes of dryer time and you will use only one dryer. The amount of time that the dryer gives you for each quarter is independent.

a. Draw a picture of the probability density function for this situation and write the formula for this density function. (8 pts)

b. What is the average length of time the dryer will run on a quarter? (4 pts

c. What is the maximum probability that you will need more than three quarters to dry your pile of laundry? (8 pts)

Answer 1