Review Problems

Review Problems Discussed in Class

When a certain machine is calibrated correctly, 10% of its output does not meet specifications. When it is not calibrated correctly, 30% of its output does not meet specifications. From past experience we know that it is calibrated correctly 60% of the time. If we draw one item at random and find that it doesn't meet specifications, what is the probability that the machine isn't calibrated correctly? If we draw a sample of 5 items and find that 2 of them don't meet specifications, what is the probability that the machine isn't calibrated correctly?

A company produces 20% of the windshields for a certain model of car at Plant A and 80% at Plant B. The mean number of flaws (small bubbles) per windshield is 2.1 at Plant A and 4.3 at Plant B. If a given windshield has 3 flaws, what is the probability that it was produced at Plant A? State any assumptions you make. [December 2005 exam]
If your time to run a marathon has a mean of 4 hr 10 min with a standard deviation of 15 min, and your friend's time has a mean of 3 hr 55 min with a standard deviation of 10 min, and you both run in the same race, what is the probability that you will finish ahead of your friend? State any assumptions you make and discuss their validity in this example. [December 2001 exam] Do a sensitivity analysis to show how this probability changes if your times are positively correlated.
If the diameter of the hole in a washer is N(10, 1) and the diameter of the shaft of a pin is N(9, 1), and a pin and a washer are selcted at random and independently of each other, what is the probabilty that the pin will fit into the washer? How will this probability change if an attempt is made to match large washers with large pins and small washers with small pins, so that the correlation coefficient is 0.6?

Suppose that the average weight of people is 66 kg and the standard deviation is 10 kg. If an elevator can take a maximum load of 750 kg, and people get on independently of each other, and we want the probability of overload to be less than 0.1%, how many people can be allowed on at one time? State any assumptions you make. Why would your calculation be invalid if you knew that there were "weight-watchers" meetings in the building every day? [T02 2005]