This assignment is relatively short. It doesn’t cover all the concepts in the Lecture; not all the concepts lent themselves well to assignment questions that I could think of. You should make sure that you read through the Lecture carefully, if you haven’t already.
Since the assignment is short, I am going to recommend that everyone do ALL problems. None of them are “challenge” problems; they all concern core concepts.
The problems are mostly taken from Chapter VII problems (page 194); I’ve added one additional problem.
Problems
All problem numbers refer to Chapter VII problems (page 194). All problems are from there, except for one problem I have written below.
Basic problems on normal approximation to binomial
Problems 3 and 4 are the most straightforward examples of applying the normal approximation to the binomial distribution. Everyone should make sure that they can solve these two problems.
a twist on normal approximation to binomial
Problem 5 is the same set-up—approximating a binomial distribution with a normal distribution—but it changes what is known and what you are looking for.
sampling
The following problem is similar to Section 7 of the Lecture, particularly to Exercise 2.
Problem: Suppose that we are polling people about which party they support. For simplicity, I will assume that there are two choices only. I will also assume that the support is expected to be not too far from 50-50. The actual proportion of the population supporting party A I will call p, and the proportion supporting party B I will call q. Then q=1-p. (I’m assuming there are only two options, and everyone has to choose. You can analyze the situation with more options basically the same way, but I will stick with only two options to keep things simple.)
Now, I poll n people selected randomly (a “sample”). For each person I select, they will support party A with probability p, and party B with probability q. This can therefore be modeled as a binomial distribution, with n experiments and a probability p of “success” each time. In the end I will find k out of n people in my sample support party A, so I will estimate the proportion of support of party A to be k/n.
Usually, k/n will not exactly equal p. This is called an “error due to sampling”. I would like to try to keep this error small. The way to do this, in theory, is to make n large enough. (In practice, you also have to worry about how truly random your sample selection is, how good your response rate is, how you phrase your questions, etc.)
Even with a large n, there will still be some probability that we are unlucky, and that we get a sample for which k/n is quite different from p. So, the best we can do is to try to minimize the probability of such an unhappy accident.
Suppose that we want to be 95% certain that our value of k/n (measured support for party A) is within 1% of the correct value p (actual support in the population for party A). (We would then say “support for party A is XX%, plus or minus 1% with 95% confidence”.)
How large does n have to be in order to achieve this?
Problem 6 is also about sampling. The wording of this problem might be a little bit confusing, but it is basically the same as the problem I asked above.
relative standard deviation of binomial
Problem 7 is about the qualitative point that I was talking about in Section 7 of the Lecture. You can answer this as a qualitative question: figure out how many standard deviations 5400 is away from the expected number of heads. How likely is the number of heads to be that many standard deviations away from the mean? (You can also calculate an exact answer for the probability of getting 5400 or more heads, and judge whether that is a likely outcome with a fair coin. For this problem though, you should be able to immediately see from the z-score, what that probability is going to be like, roughly speaking.)
