IV. Experience with Random Behavior
The study of randomness, probability, and inference in ActivStats takes advantage of the computer as a platform for experiment and experience. Students discover basic concepts including probability, the law of large numbers, the central limit theorem, sampling distributions, confidence intervals, and the reasoning of hypothesis testing. Lessons 12–15 fit together closely. Students will lose much of the benefit of this approach if lessons are skipped or done out of order.
12. Randomness
We introduce a tool that generates truly random outcomes of a color based on the random selection of a location along a rectangular bar that is colored red and blue. The choice of color under the selector depends upon the student's reaction time rather than on a pseudorandom generator, and is thus truly random.
With the help of the tool, we learn that
Randomness consists of short-run unpredictability and long-run regularity.
A video shows basketball shooting as random in this sense. The random simulation tool graphs the developing proportion of red outcomes so students can watch the graph settle down. In a later activity, the bars in the tools are blank so that students learn that they can estimate an unknown quantity by collecting many random outcomes.
Teacher's Notes: The tool used for interaction here (and in succeeding lessons) has several important properties:
This tool and others that generalize it will support much of the learning for probability and inference, so it is important for students to understand and be comfortable with the way that it works.
Although ActivStats does not pursue this idea, one could introduce a discussion of where the randomness is. For example, at what point in speeding up the selector and adding more and more thin colored stripes did the outcome become random? This chain of reasoning is philosophically deep, but it often intrigues students.
If you do not plan to teach the optional Conditional Probability lesson, you may want to direct students to the third page of that lesson to view the "Hot Hand" video, which is always good for a class discussion about randomness.
Technical Notes: Many of the examples generated in the expositions are themselves random and will be different for each student and in successive viewings by the same student. Likewise the true proportion red in the bars is randomly generated for each example. When multiple bars are present (late in the lesson), they are started at (pseudo-) random locations (to thwart attempts to "nudge" the selectors) and given (pseudo-) random speeds (to make the results independent), but they do honestly stop when the mouse button is raised.
This lesson is very much a computer-centered one, and does not correspond to material in most texts, although it is compatible with virtually any statistics text.
Students who are color blind and find it difficult to differentiate between red and blue on the screen can change the colors displayed with a mouse click on the color bins. The narrator, of course, will continue to call these colors "red" and "blue".
13. Intuitive Probability
Extending the work with random phenomena, we say that...
Probability is the name we give to the long-run regularities found in random phenomena.
What we saw as estimating the fraction of the bar that was red could be seen as estimating the probability of a red outcome. We introduce the basic laws of probability.
We generalize to multinomial outcomes by introducing more than 2 colors and discover that the proportion of outcomes of each color settles down in the long run.
We discuss and demonstrate the Addition Rule for disjoint events and the Multiplication rule for independent events. A new tool, that starts out looking much like the original randomness tool, introduces a new kind of outcome; shading in the bar. This tool shows students both outcomes that are independent and outcomes that are not independent.
Teacher's Notes: Our approach to probability recalls the use of the terms "relative frequency" and "proportion" from earlier lessons and by relating the area in the bars with a specific color to probability, builds on the area principle and the definition of a density curve. The goal is to provide students with enough probability to work with inference, but to base their understanding on intuitively natural concepts.
The tool that generates shaded as well as colored results is used again in the conditional probability chapter and then again for discussing contingency tables and chi square.
14. Conditional Probability
An example about how the probability of a low birthweight baby might depend upon whether the mother smokes introduces the idea of conditional probability.
When we restrict our attention to cases that satisfy a specified condition, the resulting probabilities are called conditional probabilities. Conditioning can be thought of as limiting the "who" of a study to a subgroup.
The new conditional probability tool lets us generate outcomes that are either shaded or not and are colored according to 2 or more colors. The outcome circles literally "fall" into cells of a two-way table.
This tool shows students that conditioning restricts attention to a single row or column of that table. From this they can conclude that to find probabilities they should Pide by the row (or column) margin total rather than by the Total number of outcomes:
Pr(Red|Shaded) = Pr(Red)/Pr(Shaded)
A stop at Data Desk notes that tables of percentages by row or column give the information needed for conditional probability.
ActivStats returns to basketball to consider a video about the "hot hand"; whether players get "hot" and make or miss a series of shots. We can phrase the question as asking whether the probability of making a shot changes, conditional on whether a player has made the previous shot (or shots). The interesting result that there is no evidence in data of any hot hand effect surprises virtually all basketball fans and players.
Teacher's Notes: This Lesson is optional. It is not needed for any subsequent lesson (but highly recommended if you plan to teach chi-square for contingency tables), but is included to complete the requirements for many curricula, including the Advanced Placement Statistics exam.
Conditional probability is unintuitive for most people—even those who study it, so there can be real benefit in showing students how to reason systematically about conditional outcomes.
The "hot hand" video is usually good for a long and heated class discussion on the nature of independence. Many basketball fans consider the hot hand to be so obvious that the suggestion that the probability of making a basket is independent of previous performance upsets them. Be warned: it can be difficult to control this discussion in some classes. And at least one student will stop you after class to press the case for the hot hand still further. The hot hand simulation that follows the video may help calm the discussion and can be used in class.
15. Random Variables
ActivStats introduces the terminology and notation of discrete and continuous random variables.
We review the difference between the sample and population and introduce the difference between sample statistics and population parameters. We want to draw conclusions about a parameter, a fixed number that describes the population. To do this, we use a statistic, calculated from a sample and subject to variation in repeated sampling from the same population.
A parameter is a number that describes an aspect or attribute of a population. A statistic is a value computed from data without using any unknown parameters.
We introduce the idea that a random outcome of the randomness tool might be, not a discrete color, but a continuous value within a range. We discuss the probability of being in an interval and learn that we can estimate the center of the range with accumulated random outcomes.
Now that we have values to average, we discover and name the Law of Large Numbers.
The Law of Large Numbers tells us that the average of many outcomes will settle down to the true mean.
We then realize that the rectangular bar is a Uniform density and that the same ideas work for other densities. In particular, we see the Normal density again, but we also see a bimodal and a skewed density. We discover that the law of large numbers works for all of these. Students try to estimate the center of the underlying densities from the accumulated mean of randomly generated values, and learn that they can do fairly well.
Teacher's Notes: This lesson is true discovery learning. The Law of Large Numbers is ideal for such discovery because it is intuitively pleasing, but commonly understood only in vague terms. The lesson works best when you sit back and let the students discover the LLN for themselves. Discuss the result in class only after everyone has had a chance to do the lessons. Working with other underlying distributions presages a similar idea for the Central Limit Theorem.
The LLN is stated in terms of the means of continuous random variables. We deliberately avoid the "fudge" of finding the LLN for proportions.
The LLN is the first real discovery that students can attribute to simulation. This is not a bad place to discuss simulation and what we can learn from it. If we did not have a tool that allowed us to work with random outcomes, we could not have found the LLN this way.
Some may find it strange that population parameters and their relationship to sample statistics have not been discussed before this point, but there is no need for these concepts earlier than this. If you are teaching from a text in which they appear in the first chapter, no harm is done, but students will probably have forgotten about them by this point in the course and will need to be reminded.
It is worthwhile to be sure that students understand the simulation and how it relates to the terms we are discussing. In particular, the population is shown in the small densities from which we sample. The sample is the set of numbers (shown in the boxes next to the density curves) gathered in the bin as a result of running the cursors and stopping them at random. The statistic is the value (usually the mean) computed from the sample and shown beneath the gathered sample values in the bin. The graph tracks the history of the mean as more and more samples are drawn.
16. Sampling Distributions
A tool lets students draw samples of sizes between 1 and 36 from each of the densities that they saw in the probability tool (and with the same mechanism as that tool). Now, however, instead of accumulating a single increasingly large sample, we make a histogram of the averages of each of successive samples, thereby graphing an approximate sampling distribution.
The histogram of the means behaves like the histogram tool of the early lessons. In particular, we can highlight the central 38% to visualize its standard deviation.
Working with this tool, we address the question of how averaging reduces variability. We know from the Law of Large Numbers that means settle down as the sample size grows. We now can estimate the standard deviation of the means of samples of different sizes. Working with this tool students discover that the standard deviation of the means of n values is reduced by 1/n.
Students observe that the histogram is Normal when we sample from the Normal. They then try other population densities and discover that even for modest sample sizes, the histogram of means of samples drawn from them looks Normal, and that this tendency grows with the sample size. This tendency is called the Central Limit Theorem.
Teacher's Notes: This chapter may be the most difficult part of the course, and probably will require more than one class. Sampling distributions are more intuitive when students can see them grow in simulations. Allowing students to discover the 1/n property and the Central Limit Theorem for themselves helps a great deal. However, the CLT is not intuitive. We try to reassure students that the CLT is unexpected for everyone at first; it is probably a good idea to do so in class as well.
One project (found under the PROJ icon in the toolbar) shows how to set the tool to simulate the sampling distribution for each of the summary statistics we learned about earlier. The exercise is worthwhile to drive home the idea of a sampling distribution and to show statistics with non-normal sampling distributions. It can be assigned as homework or made the center of a class project (if sufficient computer equipment is available.)
Background Notes: Although ActivStats emphasizes the central importance of randomization in generating sampling distributions this point is often missed and is well worth repeating. Some texts leave the impression that sampling distributions arise either from measurement error or magically, This is a good place to ensure that students understand this fundamental concept.