One of the key ideas in statistics is that sometimes we will be wrong. When we report a 95% confidence interval, we will be wrong 5% of the time. Or in other words, about 1 in 20 of 95% confidence intervals will not contain the population parameter we are attempting to estimate. That is how they are defined. The thing is, we always think we are part of the 95% rather than the 5%. Mostly we will be correct, but if we do enough statistical analysis, we will almost definitely be wrong at some point. However, human nature is such that we tend to think it will be someone else. There is also a feeling of blame associated with being wrong. The feeling is that if we have somehow missed the true value with our confidence interval, it must be because we have made a mistake. However, this is not true. In fact we MUST be wrong about 5% of the time, or our interval is too big, and not really a 95% confidence interval.
The term “margin of error” appears with increasing regularity as elections approach and polling companies are keen to make money out of sooth-saying. The common meaning of the margin of error is half the width of a 95% confidence interval. So if we say the margin of error is 3%, then about one time in twenty, the true value of the proportion will actually be more than 3% away from the reported sample value.
What doesn’t help is that we seldom do know if we are correct or not. If we knew the real population value we wouldn’t be estimating it. We can contrive situations where we do know the population but pretend we don’t. If we do this in our teaching, we need to be very careful to point out that this doesn’t normally happen, but does in “classroom world” only. (Thanks to MD for this useful term.) General elections can give us an idea of being right or wrong after the event, but even then the problem of non-sampling error is conflated with sampling error. When opinion polls turn out to miss the mark, we tend to think of the cause as being due to poor sampling, or people changing their minds, or all number of imaginative explanations rather than simple, unavoidable sampling error.
So how do we teach this in such a way that it goes beyond school learning and is internalised for future use as efficient citizens?
I have two suggestions. The first is a series of True/False statements that can be used in a number of ways. I have them as part of on-line assessment, so that the students are challenged by them regularly. They could be well used in the classroom as part of a warm-up exercise at the start of a lesson. Students can write their answers down or vote using hands.
Here are some examples of True/False statements (some of which could lead to discussion):
You can check your answers at the end of this post.
The other teaching suggestion is for an experiential exercise. It requires a little set up time.
Make a set of cards for students with numbers on them that correspond to the point estimate of a proportion, or a score that will lead to that. (Specifications for a set of 35 cards representing the results from a proportion of 0.54 and 25 trials is given below).
Introduce the exercise as follows:
“I have a computer game, and have set the ratio of wins to losses at a certain value. Each of you has played 25 times, and the number of wins you have obtained will be on your card. It is really important that you don’t look at other people’s cards.”
Hand them out to the students. (If you have fewer than 35 in your class, it might be a good idea to make sure you include the cards with 8 and 19 in the set you use – sometimes it is ok to fudge slightly to teach a point.)
“Without getting information from anyone else, write down your best estimate of the true proportion of wins to losses in the game. Do you think you are correct? How close do you think you are to the true value?”
They will need to divide the number of wins by 25, which should not lead to any computational errors! The point is that they really can’t know how close their estimate is to the true value – and what does “correct” mean?
Then work out the margin of error for a sample of size 25, which in this case is estimated at 20%. Get the students to calculate their 95% confidence intervals, and decide if they have the interval that contains the true population value. Get them to commit one way or the other.
Now they can talk to each other about the values they have.
There are several ways you can go from here. You can tell them what the population proportion was from which the numbers were drawn (0.54). They can then see that most of them had confidence intervals that included the true value, and some didn’t. Or you can leave them wondering, which is a better lesson about real life. Or you can do one exercise where you do tell them and one where you don’t.
This is an area where probability and statistics meet. You could make a nice little binomial distribution problem out of being correct in a number of confidence intervals. There are potential problems with independence, so you need to be a bit careful with the wording. For example: Fifteen students undertake separate statistical analyses on the topics of their choice, and construct 95% confidence intervals. What is the probability that all the confidence intervals are correct, in that they do contain the estimated population parameter? This is well modelled by a binomial distribution with n =15 and p=0.05. P(X=0)=0.46. And another interesting idea – what is the probability that two or more are incorrect? 0.17 is the answer. So there is a 17% chance that more than one of the confidence intervals does not contain the population parameter of interest.
This is an area that needs careful teaching, and I suspect that some teachers have only a sketchy understanding of the idea of confidence intervals and margins of error. It is so important to know that statistical results are meant to be wrong some of the time.
Answers: T,T,F, debatable, F,F.
Data for the 35 cards:
|Number on card|
|Number of cards|