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Chapter 3
Estimation of p
3.1 Point and Interval Estimates of p
Suppose that we have Bernoulli Trials (BT). So far, in every example I have told you the (numer- ical) value of p. In science, usually the value of p is unknown to the researcher. In such cases, scientists and statisticians use data from the BT to estimate the value of p. Note that the word estimate is a technical term that has a precise definition in this course. I don’t particularly like the choice of the word estimate for what we do, but I am not the tsar of the Statistics world! It will be very convenient for your learning if we distinguish between two creatures. First, is Nature , who knows everything and, in particular, knows the value of p. Second is the researcher who is ignorant of the value of p. Here is the idea. A researcher plans to observe n BT, but does not know the value of p. After the BT have been observed the researcher will use the information obtained to make a statement about what p might be. After observing the BT, the researcher counts the number of successes, x, in the n BT. We define pˆ = x/n, the proportion of successes in the sample, to be the point estimate of p. For example, if I observe n = 20 BT and count x = 13 successes, then my point estimate of p is pˆ = 13/20 = 0. 65. It is trivially easy to calculate pˆ = x/n; thus, based on your experiences in previous math courses, you might expect that we will move along to the next topic. But we won’t. What we do in a Statistics course is evaluate the behavior of our procedure. What does this mean? Statisticians evaluate procedures by seeing how they perform in the long run. We say that the point estimate pˆ is correct if, and only if, pˆ = p. Obviously, any honest researcher wants the point estimate to be correct. Let’s go back to the example of a researcher who observes 13 successes in 20 BT and calculates pˆ = 13/20 = 0. 65. The researcher schedules a press conference and the following exchange is recorded.
- Researcher: I know that all Americans are curious about the value of p. I am here today to announce that based on my incredible effort, wisdom and brilliance, I estimate p to be 0.65.
- Reporter: Great, but what is the actual value of p? Are you saying that p = 0. 65?
- Researcher: Well, I don’t actually know what p is, but I certainly hope that it equals 0.65. As I have stated many times, nobody is better than I at obtaining correct point estimates.
- Reporter: Granted, but is anybody worse than you at obtaining correct point estimates?
- Researcher: (Mumbling) Well, no. You see, the problem is that only Nature knows the actual value of p. No mere researcher will ever know it.
- Reporter: Then why are we here? Before we follow the reporter’s suggestion and give up, let’s see what we can learn. Let’s bring Nature into the analysis. Suppose that Nature knows that p = 0. 75. Well, Nature knows that the researcher in the above press conference has an incorrect point estimate. But let’s proceed beyond that one example. Consider a researcher who decides to observe n = 20 BT and use them to estimate p. What will happen? Well, we don’t know what will happen. The researcher might observe x = 15 successes, giving p ˆ = 15/20 = 0. 75 which would be a correct point estimate. Sadly, of course, the researcher would not know it is correct; only Nature would. Given what we were doing in Chapters 1 and 2, it occurs to us to calculate a probability. After all, we use probabilities to quantify uncertainty. So, before the researcher observes the 20 BT, Nature decides to calculate the probability that the point estimate will be correct. This probability is, of course,
P (X = 15) =
(0.75)^15 (0.25)^5 ,
which I find, with the help of the binomial website, to be 0.2023. There are two rather obvious undesirable features to this answer.
- Only Nature knows whether the point estimate is correct; indeed, before the data are col- lected, only Nature can calculate the probability the point estimate will be correct.
- The probability that the point estimate will be correct is disappointingly small.
(And note that for most values of p, it is impossible for the point estimate to be correct. For one of countless possible examples, suppose that n = 20 as in the current discussion and p = 0. 43. It is impossible to obtain pˆ = 0. 43 .) As we shall see repeatedly in this course, what often happens is that by collecting more data our procedure becomes ‘better’ in some way. Thus, suppose that the researcher plans to observe n = 100 BT, with p still equal to 0.75. The probability that the point estimate will be correct is,
P (X = 75) =
(0.75)^75 (0.25)^25 ,
which I find, with the help of the website, to be 0.0918. This is very upsetting! More data makes the probability of a correct point estimate smaller, not larger. The difficulty lies in our desire to have pˆ be exactly correct. Close is good too. In fact, statisti- cians like to say,
p = 0. 05 then the probability the interval will be correct is 0.9885, the same as it is for p = 0. 95.
- Viewed as a function of p, the probability of being correct is minimized at or near p = 0. 50 and it generally grows larger as we move away from 0.50, towards either 0 or 1.
B/c statisticians are disappointed with fixed-width interval estimates, we will turn our attention to the idea of ‘fixed probability of being correct.’ The method is described below. In Chapter 2, we saw pictures of probability histograms that suggest approximating binomial probabilities by using a normal curve. I did examples and you did homework that revealed that in many instances these approximate answers are quite good. In fact, the method works very well provided that p is not too close to 0 and 1 and that n is pretty large. At this time, we will use these admittedly extremely vague expressions ‘not too close’ and ‘pretty large.’ We will eventually deal with this issue, but not now. First, it is bothersome to keep saying ‘p is not too close to either 0 or 1.’ So we avoid this, as follows. I will assume that the researcher is a good enough scientist to distinguish between situations in which p is very close to 0 (say 0.01 or smaller) and very close to 1 (say 0.99 or larger). I really cannot imagine that a researcher would be sufficiently ignorant of the subject of study to not be able to do this! For dichotomous trials the labels of success and failure are arbitrary. In my experience it seems to be human nature to called the preferred outcome, if there is one, the success. For example, if I am shooting free throws, I call a made shot a success and a miss a failure. We will follow this practice unless we believe that one of the outcomes is unlikely; that is, either p or q is close to 0. For reasons that will become apparent later, we greatly prefer to have p near 0 than to have p near
- As a result, henceforth we will obey the following rule:
For BT, if one of the possible outcomes has probability of occurring that is be- lieved to be close to 0, we will designate that outcome as the success.
I have talked about Nature knowing the value of p and the researcher not knowing it. As a mathematician, I think about p having a continuum of possible values between 0 and 1. (Exclu- sive; remember we are not interested in BT that always give successes or always give failures.) But scientifically, unless p is very close to 0, I am happy with knowing p to, say, three digits of precision. I will give two examples. Recall that one of the most important applications of BT is when a researcher selects a random sample, with replacement, from a finite population. Consider the 2008 presidential election in Wisconsin. Barack Obama received 1,677,211 votes and John McCain received 1,262,393 votes. In this example, I will ignore votes cast for any other candidates. The population size is N = 1,677,211 + 1,262,393 = 2,939,604. I will designate a vote for Obama as a success, giving p =
- 571 and q = 0. 429. Notice I say that p = 0. 571. I conjecture that this imprecision did not bother you. In particular, you did not jump up (figuratively or literally) and say,
No! The value of p is the rational number 1,677,211 divided by 2,939,604, which as a decimal is 0.570556782.... And I apologize for not writing this decimal until it repeats, but this is the size of the display on my calculator and I have other work I must do.
Personally, and this is clearly a value judgment that you don’t need to agree with, 0.571 is precise enough for me: Obama received 57.1% of the votes. If I am feeling particularly casual, I would be happy with 0.57. I would never be happy, in an election, to round to one digit, in this case 0.6, because for so many elections rounding to one digit will give 0.5 for each candidate, which is not very helpful! (Of course, sometimes we must focus on total votes, not proportions. For example, in the 2008 Minnesota election for U.S. Senator, Franken beat Coleman by a small number of votes. The last number I heard was that Franken had 312 more votes out of nearly 3 million cast. So yes, to three digits, each man received 50.0% of the votes.) For p close to 0 (remember, we don’t let it be close to 1), usually we want much more precision than simply the nearest 0.001. At the time of this writing, there is a great deal of concern about the severity with which the H1N1 virus will hit the world during 2009–10. Let p be the proportion of, say, Americans who die from it. Now, if p equals one in 3 million, about 100 Americans will die, but if it equals one in 3,000, about 100,000 Americans will die. To the nearest 0.001, both of these p’s is 0.000. Clearly, more precision than the nearest 0.001 is needed if p is close to 0.
3.2 The Approximate 95% Confidence Interval for p
In this section we learn about a particular kind of interval estimate of p which is called the confi- dence interval (CI) estimate. I will first give you the confidence interval formula and then derive it for you. Remember, first and foremost, a confidence interval is a closed interval. An interval is determined by its two endpoints, which we will denote by l for lower (smaller) endpoint and u for upper (larger) endpoint. Thus, I need to give you the formulas for l and u. They are:
l = ˆp − 1. 96
√ p ˆq/nˆ and u = ˆp + 1. 96
√ p ˆq/n.ˆ
If you note the similarity of these equations and recall the prevalence of laziness in math, you won’t be surprised to learn that we usually combine these into one expression for the 95% confidence interval for p:
p ˆ ± 1. 96
√ p ˆq/n.ˆ
We often write this as p ˆ ± h,
with h = 1. 96
√ p ˆq/n,ˆ
called the half-width of the 95% CI for p. I will now provide a brief mathematical justification of our formula.
After some algebra, it follows that l ≤ 0. 400 corresponds to pˆ ≤ 0. 470 and u ≥ 0. 400 corresponds to pˆ ≥ 0. 340. Remembering that pˆ = x/ 200 , we conclude that the confidence interval will be correct if, and only if, 68 ≤ X ≤ 94 , where probabilities for X are given by the Bin(200,0.40). With the help of the binomial website, this probability is found to be 0.9466. Not ideal—I would prefer 0.9500—but a reasonably good approximation. I will repeat the above example for the same n = 200, but for a p that is closer to 0, say p = 0. 10. In this case, by algebra, the confidence interval is correct if, and only if, 15 ≤ X ≤ 30. The probability of this event is 0.8976, which is not very close to the desired 95%. For one last example, suppose that n = 200 and p = 0. 01. The interval is correct if, and only if, 1 ≤ X ≤ 8. The probability of this event is 0.8658, which is a really bad approximation to 0.9500. We have seen that for n = 200, if p is close to 0 the 95% in the 95% confidence interval is not a very good approximation to the exact probability that the interval will be correct. We will deal with that issue soon, but first I want to generalize the above result from 95% to other confidence levels.
3.2.1 Other Confidence Levels and One-sided Intervals
The 95% confidence level is very popular with statisticians and scientists, but it is not the only possibility. You could choose any level you want, provided that it is above 0% and below 100%. There are six levels that are most popular and we will restrict attention to those in this class. They are: 80%, 90%, 95%, 98%, 99% and 99.73%. Consider again our derivation of the 95% confidence interval. The choice of 95% for level led to 1.96 appearing in the formula, but otherwise had absolutely no impact on the algebra or probability theory used. Thus, for any other level, we just need to determine what number to use in place of 1.96. For example, for 90% we need to find a positive number, let’s call it z, so that the area under the snc between −z and +z is 90%. It can be shown that z = 1. 645 is the answer. Thus, to summarize: The 90% confidence interval for p is
p ˆ ± 1. 645
√ p ˆqˆ n
Extending these ideas we get the following result. The (two-sided) confidence interval for p is given by:
p ˆ ± z
√ p ˆqˆ n
In this formula, the number z is determined by the desired confidence level, as given in the follow- ing table.
Confidence Level 80% 90% 95% 98% 99% 99.73% z: 1.282 1.645 1.960 2.326 2.576 3.
Thus, for example,
p ˆ ± 2. 576
√ p ˆqˆ n
is the 99% two-sided confidence interval for p. Also,
p ˆ ± 3
√ p ˆqˆ n
is the 99.73% CI for p. We also recognize this as the pretty certain interval of Chapter 1. Thus, the pretty certain interval of Chapter 1 was simply the 99.73% CI for r. (Remember, in Chapter 1 the probability of interest was denoted by r.) You have no doubt noticed that I have added the modifier two-sided to the technical term confidence interval. We call our answer the two-sided CI because it has both upper and lower bounds. Sometimes in science we want a one-sided bound on the value of p. This is especially true when p is close to 0. Below are the two results. The upper confidence bound for p is given by:
p ˆ + z 1
√ p ˆqˆ n
and the lower confidence bound for p is given by:
p ˆ − z 1
√ p ˆqˆ n
In these formulas, the number z 1 is determined by the desired confidence level, as given in the following table.
Confidence Level 90% 95% 97.5% 99% 99.5% 99.86% z 1 : 1.282 1.645 1.960 2.326 2.576 3.
For example, suppose that n = 200 and pˆ = 0. 250. The 95% upper confidence bound for p is given by:
√ 0 .25(0.75) 200
In words, I am 95% confident that p is 0.300 or smaller.
3.3 Exact Confidence Intervals
In an earlier example we saw that if n = 200 and p is close to 0, our above method, based on the snc approximation, is not very good. It is not very good because the actual true probability that the 95% confidence interval will be correct is substantially smaller than 95%. There is an exact method available for obtaining a confidence interval for p. It can be obtained by using the website:
If you want, say, 90%, instead of 95%, the above is true with the number 90%. It is also true for any one-sided CI (upper bound or lower bound). To make sure this is clear: If you want, say, 95% confidence:
You can use the website which guarantees we have a 95% chance or more of getting a correct interval regardless of the value of p
OR
You can use the snc which guarantees nothing; it gives approximately 95% and some- times the approximation is bad.
The second question now is rather obvious: Why would one ever use the snc approximation? One advantage of the snc is that it actually makes sense in that we can see how it relates to the shape of the binomial distribution. By contrast, the website answer is totally mysterious. Next, for really large studies the two methods give about the same answer. For example, if n = 2000 and x = 1000 it can be shown that the exact 95% CI is 0.4778 to 0.5222, while the snc answer is 0.4781 to 0.5219. If we round these answers to the third digit after the decimal, we get [0. 478 , 0 .522] for both. The exact answer involves some pretty serious computations, but the snc approximate answer can be obtained easily with a hand calculator. Finally, as a statistician, I feel that I understand pretty well the strengths and weaknesses of using the snc method. I don’t know who wrote the program for the website that gives exact CI’s and although it seems ok to me, I don’t really know that. I do not recommend you believe every- thing you find on a website. (Nor should you automatically believe everything anyone tells you, including me.)
