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Let us say that you are taking AP Statistics. The prerequisite is a passing grade of D or above in Algebra II. The kids that you are working with struggle with algebra and do not retain information very well. Even though you spent a month talking about z-scores and how to find them using the invNorm() function a lot of them are still confused as to what to do and need someone to spoonfeed them with simplified information that does not contain too many technical terms. Your challenge now is to teach them confidence intervals which involve dealing with a) the concept b) the math c) the interpretation and finally the d) misconceptions. As you can see, this is an uphill battle and what worsens it is the student apathy.

You review how to find the corresponding z-scores of the middle 95th percentile using the invNorm() function. You explain that the first entry must be the area to the left however when they see $z_{\alpha}$ they think the area to the right. This has been an ongoing confusion for one month despite you repeating the same thing over weeks. Now you try giving them a motivating example: "Lets say you wanted to figure out the population mean length of all the world's bald eagles' wingspan. This is our parameter. Remember a parameter is a numerical value assigned to a whole population. You take a sample of 100 bald eagles and measure their wingspan. We call this sample mean a point estimate. Do you guys think that this point estimate accurately reflects the population mean?" (Introducing vocabulary) You transition into talking about 95% confidence intervals conceptually. They are lost. Then you present them this formula. They are now completely and hopelessly lost:

$ \bar{x} - z_{\frac{\alpha}{2}}\sigma < \mu < \bar{x} + z_{\frac{\alpha}{2}}\sigma $

You draw a picture showing that the level of confidence is $1-\alpha$ and the remaining areas we don't want are $\frac{\alpha}{2}$ and $\frac{\alpha}{2}$. They don't get what's going on. Twenty minutes in, there are some students not paying attention anymore and doing another class' work.

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Advanced Placement and it is a way for high school students to earn college credit by taking a test at the end of the year. –  John Habert Feb 6 at 19:57
    
sorry to be clueless about the US system, but how old are these kids? And is college where they go at 18 same as university for us Brits? –  TooTone Feb 6 at 20:03
    
I think that the prereq of Alg II is not high enough. I admit that algII pretty much covers enough algebra to work and manipulate with the stats formula, but the word problems associated with AP stats requires a different thinking skill. An algII student in my view hasn't quite yet acquired this kind of thinking and reasoning. If you want your AP stats class to be more than plugging numbers into equations, then the prereq needs to be increased. Interpretation of the result is just as important as getting the numbers. –  imranfat Feb 6 at 20:13
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Judging from your posting history, you don't appear to be an educator, but a student. You have asked multiple questions about statistical inference at an introductory level, being careful to frame some questions as if you were asking how to explain a concept from an educator's perspective, but you have not answered any other questions yourself, and you have rarely confirmed an answer to your questions. Other questions you've asked indicate you are unlikely to have the necessary background to be a teacher of statistics or mathematics. –  heropup Feb 7 at 4:45
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In light of the above, I find your behavior to be less than honest, and I suspect you are really just asking to be taught statistics. In that case, rather than expecting the math.SE community to write you an entire textbook on introductory statistics, perhaps you should actually read any one of the multitude of excellent introductory texts on the subject, and stop expecting others to spoon-feed you. –  heropup Feb 7 at 4:48

3 Answers 3

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I'm going to concentrate on the concept, the interpretation and misconceptions. I believe all of these can be well addressed by taking an interactive approach. The math I believe should come later after the concepts are established, certainly at an introductory level, and in any case this isn't the place to rehearse all the maths required to explain confidence intervals.

Essentially what I think you can do is find a large population (or large sample) of data, randomly sample from it and produce a confidence interval. If you can't find such a sample, just simulate it. That's what I've done below, simulating taking samples of size 20 and constructing 90% confidence intervals from a normal distribution with mean 100 and standard deviation 10.

After 5 experiments I got this

enter image description here

You can see the first, blue confidence interval doesn't contain the mean, but the rest do. So I was "unlucky" the first time, and "lucky" the remaining times. With a simple approach such as this you can discuss things like

  • concept: producing a confidence interval is like running an experiment. If you repeat the experiment enough then a p% confidence interval will contain the mean p% of the time.
  • interpretation: the higher the confidence interval, the more plausible it is that the true mean is contained within the interval. But if you reject the hypothesis that the true mean is within the interval, you will be wrong (100-p)% if the time.
  • misconceptions: one misconception is that given a confidence interval there's a 95% chance that the given confidence interval contains the true mean. That's false, as you can see with the picture above: for all the confidence intervals shown the confidence intervals contain the mean with 0 and 1 probability. The mean is fixed, and when you run your experiment and sample data, there's a 95% chance that you'll sample "good" data that contains the mean.

Other than that, an interactive approach is good because it means you can ask questions like

  • does the interval get smaller or larger if you increase the confidence level?
  • what happens if you increase the sample size?
  • (advanced) what happens if you change the underlying distribution from which you sample?

I used this approach with a free and widely used software package called R, available on all the main operating systems. The code is below

#SETUP CODE -- RUN ONCE
#20 items per random sample
n=20
#set up empty matrix to hold data
data=matrix(ncol=0, nrow=n)

#SAMPLING AND PLOTTING BELOW -- RUN ONCE PER EXPERIMENT
#randomly choose data from normal distribution and add to data matrix
r=rnorm(n, 100, 10)
data=cbind(data, r)

#plot data and confidence intervals
conf=90
num_means=dim(data)[2]
plot(x=c(0,100), y=c(100,100), type="l", ylim=c(80,120), xlim=c(1,num_means +0.5), ylab="random sample", xlab="experiment number")
for(i in 1:num_means) {
points(x=rep(i, n), y=data[,i])
se=sd(data[,i])/sqrt(n)
conf_interval=c(mean(data[,i])+qt((100-conf)/200, n-1)*se,mean(data[,i])+qt(1-(100-conf)/200, n-1)*se)
if (conf_interval[1] <= 100 & 100 <= conf_interval[2]) COL= "red" else COL= "blue"
points(x=c(i+0.1,i+0.1), y=conf_interval, type="l", lwd=2, col=COL) }

You might try purchasing a book on introductory statistics with R and working through the explanation and examples. There are plenty of books which teach statistics and use a software package along the way.

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I guess I'd have to ask WHY weak math students are even taking AP Stats?!! That aside, I would explain confidence intervals intuitively (if that's possible to say about confidence intervals) by using a "magic 8 ball" or "weather forecasting" analogy (perhaps I'm dating myself here! ;-P).

An x% confidence interval is like an oracle or weather forecaster that is right x% of the time. However, for any particualr sample, you don't know if it contains the mean or not, but you trust that it does becasue it is right X% of the time (with X being reasonably hight, like the canonical 95%), just like if a weather foreacaster predicts rain, and they have a track record of being right 95% of the time, then you'd be well served to take an umbrella!

I wouldn't get too into the whole issue of frequentist vs bayesian interpretations of probability, as even professional statisticians argue about that stuff. Particularly grating is the annoying "You can't say that there is a 95% probability that a particular 95% CI contains the true mean"...YES that is true in an technical, and solely frequentist interpretation of probability, as the true mean is a fixed number.

However, the word "confidence" is closer to an average person's notion of probability than the technical frequency-based sense, as in "I am 95% certain that we will win the match tomorrow". To most people, I think the frequentist sense of probability only gets its intuitive strength by being connected to this subjective sense of certianty.

Anyway, if you students are glazing over at the basics above, then perhaps try something like the above. Good luck - hope your student's benefit from my and other's input.

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I can partially answer why weak math students are taking AP Stats - it is the most attractive option to them. Most colleges want 4 years of math. This means Algebra I, Geometry, Algebra II and a 4th. Since there is little algebra and a lot of calculator work in AP Stat compared to precalculus, plus a chance to get some college credit, many students go the AP route. Plus any AP class looks good to colleges. –  John Habert Feb 7 at 20:08
    
@JohnHabert I guess I can see the appeal then...although a D+ and a 2 on the AP exam won't help many people. –  Eupraxis1981 Feb 8 at 4:58

Confidence intervals are what they sound like: How certain you are an interval contains the true mean. Of course you can be more certain by widening your area, like "There is a high probability eagles' wingspan is within 0 cm and 1000 cm" but you will be lowering your accuracy. The same way you can increase accuracy by lowering how sure you are. It's important especially for AP that you describe the difference between a sample, sample mean and population mean. Unless you measure an entire population you will never know for certain the true mean, but we can get a percentage to how likely our interval has the true number within it. Don't overload on vocabulary and you should understand z scores and the normal distribution before talking about this.

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