Copyright © Cengage Learning. All rights reserved. 11 Applications of Chi-Square

Copyright © Cengage Learning. All rights reserved Chi-Square Statistic

3 Cooling a Great Hot Taste

4 If you like hot foods, you probably have a favorite hot sauce and preferred way to “cool” your mouth after eating a mind-blowing spicy morsel. Some of the more common methods used by people are drinking water, milk, soda, or beer or eating bread or other food. There are even a few who prefer not to cool their mouth on such occasions and therefore do nothing. Source: Data from Anne R. Carey and Suzy Parker, © 1995 USA Today.

5 Cooling a Great Hot Taste Recently a sample of two hundred adults professing to love hot spicy food were asked to name their favorite way to cool their mouth after eating food with hot sauce. The table summarizes the responses. [EX11- 01] Count data like these are often referred to as enumerative data.

6 Cooling a Great Hot Taste There are many problems for which enumerative data are categorized and the results shown by way of counts. For example, a set of final exam scores can be displayed as a frequency distribution. These frequency numbers are counts, the number of data that fall in each cell. A survey asks voters whether they are registered as Republican, Democrat, or Other, and whether or not they support a particular candidate. The results are usually displayed on a chart that shows the number of voters in each possible category.

7 Data Set-Up

8 Suppose that we have a number of cells into which n observations have been sorted. (The term cell is synonymous with the term class; the terms class and frequency were defined and first used in earlier chapters. Before you continue, a brief review of Sections 2.1, 2.2, and 3.1 might be beneficial.)

9 Data Set-Up The observed frequencies in each cell are denoted by O 1, O 2, O 3,..., O k (see Table 11.1). Note that the sum of all the observed frequencies is O 1 + O O k = n where n is the sample size. Table 11.1 Observed Frequencies

10 Data Set-Up What we would like to do is compare the observed frequencies with some expected, or theoretical, frequencies, denoted by E 1, E 2, E 3,..., E k (see Table 11.1), for each of these cells. Again, the sum of these expected frequencies must be exactly n: E 1 + E E k = n

11 Data Set-Up We will then decide whether the observed frequencies seem to agree or disagree with the expected frequencies. We will do this by using a hypothesis test with chi- square, χ 2 (“ki-square”; that’s “ki” as in kite; χ is the Greek lowercase letter chi).

12 Outline of Test Procedure

13 Outline of Test Procedure In repeated sampling, the calculated value of χ 2 in formula (11.1) will have a sampling distribution that can be approximated by the chi-square probability distribution when n is large. This approximation is generally considered adequate when all the expected frequencies are equal to or greater than 5.

14 Outline of Test Procedure We know that the chi-square distributions, like Student’s t-distributions, are a family of probability distributions, each one being identified by the parameter number of degrees of freedom, df. The appropriate value of df will be described with each specific test. Assumption for using chi-square to make inferences based on enumerative data The sample information is obtained using a random sample drawn from a population in which each individual is classified according to the categorical variable(s) involved in the test.

15 Outline of Test Procedure A categorical variable is a variable that classifies or categorizes each individual into exactly one of several cells or classes; these cells or classes are all-inclusive and mutually exclusive. The side facing up on a rolled die is a categorical variable: the list of outcomes {1, 2, 3, 4, 5, 6} is a set of all-inclusive and mutually exclusive categories.