AP Statistics Section 14.2 A

The two-sample z procedures of chapter 13 allowed us to compare the proportions of successes in two groups (either two populations or two treatment groups in an experiment). We need a new statistical test if we want to compare more than two groups.

A contingency table (or two-way frequency table) is a table in which frequencies correspond to two variables. One variable categorizes rows and the other columns. Discussed earlier in section 4.2.

Example 14.1: Market researchers know that background music can influence the mood and purchasing behavior of customers. One study in a supermarket in Northern Ireland compared three treatments: no music, French accordion music and Italian string music. Under each condition, the researchers recorded the numbers of bottles of French, Italian and other wine purchased. Here is a table that summarizes the data:

Music NoneFrenchItalianTotal French Wine Italian Chosen Other Total

Example: Calculate the conditional distribution (in proportions) of wine chosen for each music type. No music playing: French music playing: Italian music playing:

The types of wine chosen seems to differ considerably across the three music treatments. The key question of course is this: “Are the differences due to random variation or are the differences statistically significant?”

Section 14.2 presents two types of hypothesis testing based on contingency tables. Tests of homogeneity are used to determine whether different populations have the same proportion of some characteristic. Tests of independence are used to determine whether a contingency table’s row variable is independent of its column variable.

While both types of tests use the same basic methods from section 14.1, the questions these two tests answer are different. A test for homogeneity tests whether the distribution of a categorical variable is the same for each of several populations or treatments. A test for independence tests whether two categorical variables are associated in some population of interest.

Test Statistic: where E = The degrees of freedom equal ___________________________ Conditions: Data must come from independent SRS’s of the populations of interest. All expected cell counts are greater than 1 and no more than 20% are less than 5

Use a test to compare the distribution of wines selected for each type of music.

Hypothesis:

Conditions:

Calculations:

Conclusions: Very difficult to word interpretation of p-value, so