Classroom demonstrations/activities/investigations

Birthday Matches:
Small class: Write months on the board, let students go up and write their birthdays under the month.  If no matches, use a RN table to generate more birthdays (or use students’ mothers birthdays) and continue. Large class: Define a section of size 60-90, go to each student, and call out loud his/her birthday to see if there is a match. You should get one fairly quickly, impressing your class - 'And now for my next feat....'.  With 22 people, there is already a .5 chance of a match.  Afterwards, you can give them an assignment using the Java applet that simulates the birthdays.  Of course, you can also derive the probability mathematically.  Probabilities can be surprising.

Drawing random samples and sampling distributions:
Give everyone a number in some systematic fashion. Using a RN table, draw a SRS from the class (perhaps let the class help you do this).  Estimate a proportion (e.g. proportion of females), for which you know the population value (or can determine it quickly by surveying your class). Repeat a number of times, and tabulate the results.  Note the possible, smallest, biggest errors of estimation, the shape of distribution, the mean and standard deviation of the distribution.  Is the biggest error observed about what you would expect from a bit of theory (i.e. 2 stdevs from the mean)?   Discuss other possible ways to sample students in the class (random/biased).  Would you expect the sampling distribution to stay exactly the same?  How might it change?  If your random samples had been larger, how do you think the distribution would have changed?   [Note that for a small class relative to sample size, the stdev of the distribution is actully a bit smaller than predicted by the usual theory]

P-values:
For Wilcoxon rank sum test, you can illustrate that confusing creature called a p-value, with a simple null model – drawing cards from a deck.  Write down say samples of size 4 and 5, calculate rank sum of smaller sample - suppose you get 14.  Ask class how to estimate the p-value using a deck with 9 cards (1-9), and lots of time (marooned on desert isle, stuck in jail,…).  Give out 9 card decks to as many people as possible.  Have them shuffle, deal out 4 cards, sum.  Poll to see how many get sums <= 14, and then repeat several times.  Calculate the proportion of deals resulting in sum <=14, double for estimated p-value.

Coin spinning:
If desk surfaces allow, pass around coins of same type/year.  Let students spin and record heads and tails.  Tally class results.  Calculate CI for true probability.   For older Canadian nickels and U.S. pennies, the probability is not close to 0.5.  Or let students take coin(s) home, and later accumulate the results in class.

Sampling beans:
Bring a big bag or jar of jelly beans/marbles/etc of at least two colours.  Have a student draw say 9.  Estimate the proportion of red beans in the jar.  Are any assumptions being made?  [Is it possible to draw a random sample? You might also relate this to the 1970 American draft lottery bias or to the problem of well-shuffling up a deck of cards - discussed by Diaconis]  How far off the mark might the estimate be? Repeat draws, and tabulate distribution.  Interesting shape?  Guess the true proportion now.  Now reveal the true proportion of red beans (make it in vicinity of .5).  Does the size of the jar matter?  How would the distribution change if sampling 36 beans?

Quincunx demonstrations:
bring a quincunx to class (purchasable here).  Ask students to come up and drop a bead or two.  Discuss the distribution.  It is Binomial, but more importantly, you are witnessing the Central Limit Theorem in action.  Discuss if what they are seeing bears any relationship to a person’s height.