Explain to students that you’ve filled each bag with 30 beads of five different colors. Their job is to determine if you did so randomly or with a bias. Check students: Which explanation is the null hypothesis? (Answer: The null hypothesis is that they were distributed randomly, with no bias.)
Ask: if the null hypothesis (no bias) is true, what would you expect the distribution of the beads to be? (Answer: I’d expect with 30 beads and 5 colors that there would be 6 of each.)
How might a scientist/statistician determine how far away the actual (observed) numbers are from the expected? Allow students time to brainstorm potential mathematical solutions. They should be able to come up with subtraction as the way of measuring difference.
The equation for chi-squared is:
x2 = [SUM] (o – e)2 / e
o = observed (this would be how many of each category there actually are)
e = expected (this is the amount predicted by random chance)
The difference represents how far each value is from the prediction of the H0.
The differences are squared to eliminate negative numbers. Then they are divided by the expected value. This is like taking an average.
The resulting values are all added up (denoted by the sum, or sigma on handout). The larger the differences, the larger the chi-squared value. Note that there is no consideration of sample size here, so a larger sample will in general mean a larger chi-squared value.
Explain: Chi-squared is a value that measures how far the observed is from the expected. If it is far enough away, we can reject the null hypothesis. In other words, we can say in this case that you, the teacher, did not act randomly. Instead, for some reason, you selected which beads to give to each group and therefore acted with bias. Naturally, this test cannot ascertain where this bias comes from. Ask students to come up with their own alternative hypotheses. Examples may include:
1.) The teacher prefers to use a certain color bead over other color beads.
2.) The teacher choice was random, but the original source of beads did not have equal numbers of each color.
Have students follow the directions on the handout to test their own bag. The will need to calculate degrees of freedom (number of colors – 1) and be taught to use the p-value chart. All tables and charts are provided on the worksheet.