Hints for Homework Assignment 4

General Remarks

Solution methods: You must do these problems rigorously, as indicated under "Instructions" on the problem sheet; i.e., if you use a S/F trial model you need to say what trial means, what success means, what n and p are, etc. Simply writing down numerical calculations is not sufficient. All problems require you to use an approximation (either normal or Poisson) rather than exact formulas involving binomial probabilities. You might be able to do some of the problems using binomial probabilities if you have a sufficiently powerful calculator, but this would defeat the purpose of these problems, and you will not get credit if you do that.

Normal approximation: Refer to the handout distributed in class. The formulas given there are the only ones you need to do the problems requiring normal approximation, and you should memorize these formulas. Be sure to also study the remarks on the this handout. Many problems are similar to problems that are worked out in class. Note that there are two formulas for normal approximation, depending on the type of probability you want to approximate.

Normal table: Even if you have a calculator with a built-in Phi function, it would be a good idea to use the table instead of the calculator, since you will need to use the table anyway in the exam. If you obtain a value that is out of range for this table (such as Phi(4)), this is almost always an indication that you did something wrong, such as applying the normal approximation where it was not appropriate. The Phi function is still defined for these values, but it is extremely close to 1 (at large positive values), or extremely close to 0 (at large negative values).

Specific Comments

Problem 1: This was an exam problem, so you can see how you would have done. In an exam you would have about 10 minutes for this problem and, of course, you wouldn't have access to notes, books, or calculators. The problem isn't particularly hard and doesn't involve any calculations. It simply tests your knowledge of the formulas for the binomial distribution and the Poisson and normal approximations, and the conditions under which each of the approximations apply. (The numbers have been chosen such that it is clear which approximation is appropriate.)
Problem 3: This is a two stage problem. The first stage should be done by normal approximation; the computation for the second stage does not require an approximation. (If you do use a suitable approximation for the second stage you get an answer that is off by about 5 % .)
Problems 4 and 6: These problems are of the same type as the airline overbooking problem discussed in Monday's class. Problem 6 was an exam problem; the numbers for the probabilities were chosen so as to make the calculation as simple as possible (and, in particular, can be done without calculator).
Problem 5: Draws with replacement can be modelled by S/F trials. Use a suitable approximation to compute the required probabilities.

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Last modified Mon 17 Feb 2003 01:29:47 PM CST