Practice questions for second exam. solutions
1. Past experience shows that .9% of all light bulbs produced are defective. If two bulbs are found to be defective in a sample of 100 GE bulbs, what can be said about GE bulbs relative to all light bulbs?
This is a binomial problem n =100, prob of defect = .009
We want to find out if 2 is unusual. List what could happen, what you expect to happen and put 2 in with the unusual things. expect .009*100 = .9 defectives so 0 or 1 would be typical what could happen:
0,1, | 2,3,4 . . . . . . 100
From minitab: that’s from calc menu

So, there is a 77% chance that you will see 0 or 1 defects and a 23% chance of 2 or more. At this point, 2 defects does not seem particularly unusual. So GE may be just as good as other manufacturers
3. Over the past five years the intersection of price and high streets has averaged 2.35 accidents per month. During the last two months, however, there have been 6 accidents.
What is the probability of this occurring if nothing has changed in the past six months that would affect the probability of an accident? Does it appear that the intersection has gotten more dangerous? Explain.
Poisson problem (average 2.35 per month is the give away)
We want to find out if 6 is unusual. List what could happen, what you expect to happen and put 6 in with the unusual things. expect 2.35*2 = 4.7 defectives so 4 or 5 would be typical what could happen:
0,1, 2,3,4, 5, | 6,7, . . . . . .no upper limit
From calc, probability distributions, poisson..

We get:
Again, 66% chance of 5 or fewer accidents

so there is a 34% chance of 6 or more so 6 is not unusual. The intersection may or may not be more dangerous, but this evidence is not enough to conclude that the intersection has gotten worse.
5. A bank manager wishes to provide prompt customer service for customers at the driveup window. The bank with a single drive up window can handle 10 customers in a 15minute period without significant delay. Currently the during the lunch hours (12-2) the bank averages 28 customers per hour at the drive-up window. The manager has been instructed to install a second drive up window if there are delays more than 10 percent of the time during the lunch hours. Is it necessary to install the new window? why or why not?
WooooHooo another Poisson! (average 28 per hour is the tip-off)
We want to find out how often we experience delays List what could happen and what would cause a delay what could happen in a fifteen minute period (average = 7):
0,1, 2,3,4, 5,6,7,8,9,10, | 11,12 . . . . . .no upper limit

o.k., so there are NO DELAYS slightly over 90% of the time, and delays slightly under 10 percent of the time. We don’t need a window now, but if traffic picks up at all we will.
6. A report by a prominent economist uses a binomial distribution to predict the number of trips an angler will take in a specific period. For example, if the probability of taking a trip is .04 and there are 90 days in the period, this information could be used to predict the number of trips an angler will take. Using this information, what is the probability that the angler will take more than one trip? If there are 1 million anglers, what is the expected number of trips for the group? Finally, critically evaluate (what is wrong with) this approach to modeling trips?
Binomial Problem n=90 and prob=.04
More than one is

So there is a 12 % chance of zero or one trips and a 88 percent chance of more than one trip.

The expected number of trips is .04*90 = 3.6 per angler or 3.6 million for all anglers.
O.K. this does not meet the criteria for binomial. The trips (trials) are not independent. If I go today , that reduces the chances that I will go the following day. P(fish Sunday|already fished on Saturday) < P(fish Sunday) Also I’m more likely to go on weekends etc, so the probability of a trip is not constant. Finally, I could take more than one trip per day, so a
“day” is not a yes or no trial.
This “prominent economist” is obviously a hack.
8. A work scheduler at the Royal Tea Company Ltd. must plan ahead to reduce the number of labor hours if raw materials are not available on time. Thirty percent of all shipments from India arrive late due to variation in wind weather etc. Royal Tea has a contract with one shipping company. Last month, the company got 28 shipments from the other side of the globe, but 14 of the shipments were late. One of the tea executives called the shipping company to complain. "You guys need to get your act together! You were were way over 30 percent this month!" The tea exec yells. "Calm down, the shipper replies, "These things are bound to happen occasionally." Who's right and why?
Binomial n=28 prob = .3 we expect .3*28 = 8.4 to be late
0,1,2,........,13 | 14, 15

98% chance of fewer than 14 late boats. The tea exec was right, 14 is way over 30 percent.