Type of activity: In-class, large-group

Statistical topics: populations, samples, sampling with replacement, binomial random variable, calculating binomial probabilities, hypothesis testing concepts, p-values

Time needed:  20 minutes, or more depending on the amount of discussion generated

Materials needed:

• An opaque bowl or an opaque bag
• Hershey Kisses: 70 silver-wrapped, 30 gold-wrapped
• Chalk & chalkboard, or overhead projector & transparency pen
• An overhead of a binomial probability table, or projection of Minitab onto screen

1. Prior to class, create the population by putting 70 silver-wrapped Kisses and 30 gold-wrapped Kisses in the bowl or bag.
2. Then, in class, describe the population to the students.  Tell them that each of the 100 Kisses has an opinion about Clinton's performance.  A silver Kiss approves of Clinton, while a gold Kiss disapproves of Clinton.  Do not tell the students how many of each type is contained in the population.  Only tell them that it is our job to try to draw conclusions about the population.
3. Describe the hypothesis test of interest: An avid Democratic, Clinton-supporter, such as James Carville, claims that Clinton's approval rating is as high as 90%, while a Republican, such as Newt Gingrich believes that it is much lower.  Ask the students to describe the null and alternative hypotheses in statistical notation, namely H₀: p = 0.90 versus H₁: p < 0.90.
4. Ask the students how we might test the hypothesis.  Work the students towards taking a random sample of 10 people with replacement, in which we let X = the number of people in the sample who approve of Clinton.  Then, X will take on values 0, 1, ..., 10.
5. Ask the students to argue that X is a binomial random variable:
1. The number sampled, n = 10, is fixed in advance.
2. There are two outcomes: approve or disapprove.
3. p = P(a Kiss approves) is unknown, but assumed to be 0.90, under the null hypothesis.  If we sample with replacement, we have independent trials and p is constant.  (Here, you can take an 'aside' to address "real" polls.  The population is typically much larger, U.S. population is N=260,000,000, say, a typical sample is around n=1000, and sampling is without replacement.  Hence, p=P(approve) changes, but is small enough to be ignorable.)
4. We let X = the number in the sample who approve of Clinton.
6. Now, it is time to take the random sample.  Ask a student to volunteer to record the data on the chalkboard (or overhead).  The student should create two columns, one labeled Approve (Silver) and one labeled Disapprove (Gold).  As the sample is selected, the student should tally the Kiss' opinion under the appropriate column. Then, walk around the room with the bowl in hand, and ask 10 different students to randomly select a Kiss from the population.  Make sure the students cannot see inside the bowl.  After each student selects a Kiss, the color should be noted, the opinion tallied, and the Kiss returned to the bowl before the next student selects.  Continue until all 10 Kisses are selected.
7. Ask the students what values of X support the alternative hypothesis.  Work the students towards "small values".
8. Once the random sample has been selected, count the number of approvers, 5, say.  Ask the students to define the p-value in English.  Work the students towards "how likely is that 5 (or fewer) Kisses would approve of Clinton if the true proportion of Kisses in the population who approve of Clinton is in fact 0.90?"
9. Calculate the p-value by looking up P(X<=5) on a binomial table with n=10 and p=0.90 (alternatively, use Minitab's binomial cdf).  Hence, the p-value here is 0.0016.
10. Argue that since it is unlikely (p=0.0016) that we would observe the sample we did if the null hypothesis were indeed true, we will reject the null hypothesis.  That is, there is sufficient evidence to conclude that p < 0.90.
11. If you have time, you might consider doing a second poll of n=20 Kisses for a different set of hypotheses, say H₀: p = 0.70 versus H₁: p > 0.70.  Repeat the steps as described above.  I like to do this second test so students have a good chance of seeing an example with a high p-value, and hence a test that fails to reject the null.  (Recall that the 70% of the population approves.)  Of course, if you don't reject for the first test, and don't fail to reject for this second test, then you'd have an opportunity to address Type I and Type II errors, since you know the true population value.

Can break down class into smaller groups.  Would require as many bowls of Kisses as there are groups.

Supplementary materials: None.