© aSup-2007 CHI SQUARE   1 The CHI SQUARE Statistic Tests for Goodness of Fit and Independence

© aSup-2007 CHI SQUARE   2 Preview  Color is known to affect human moods and emotion. Sitting in a pale-blue room is more calming than sitting in a bright-red room  Based on the known influence of color, Hill and Barton (2005) hypothesized that the color of uniform may influence the outcome of physical sports contest  The study does not produce a numerical score for each participant. Each participant is simply classified into two categories (winning or losing)

© aSup-2007 CHI SQUARE   3 Preview  The data consist of frequencies or proportions describing how many individuals are in each category  This study want to use a hypothesis test to evaluate data. The null hypothesis would state that color has no effect on the outcome of the contest  Statistical technique have been developed specifically to analyze and interpret data consisting of frequencies or proportions  CHI SQUARE

© aSup-2007 CHI SQUARE   4 PARAMETRIC AND NONPARAMETRIC STATISTICAL TESTS  The tests that concern parameter and require assumptions about parameter are called parametric tests  Another general characteristic of parametric tests is that they require a numerical score for each individual in the sample. In terms of measurement scales, parametric tests require data from an interval or a ratio scale

© aSup-2007 CHI SQUARE   5 PARAMETRIC AND NONPARAMETRIC STATISTICAL TESTS  Often, researcher are confronted with experimental situation that do not conform to the requirements of parametric tests. In this situations, it may not be appropriate to use a parametric test because may lead to an erroneous interpretation of the data  Fortunately, there are several hypothesis testing techniques that provide alternatives to parametric test that called nonparametric tests

© aSup-2007 CHI SQUARE   6 NONPARAMETRIC TEST  Nonparametric tests sometimes are called distribution free tests  One of the most obvious differences between parametric and nonparametric tests is the type of data they use  All the parametric tests required numerical scores. For nonparametric, the subjects are usually just classified into categories

© aSup-2007 CHI SQUARE   7 NONPARAMETRIC TEST  Notice that these classification involve measurement on nominal or ordinal scales, and they do not produce numerical values that can be used to calculate mean and variance  Nonparametric tests generally are not as sensitive as parametric test; nonparametric tests are more likely to fail in detecting a real difference between two treatments

© aSup-2007 CHI SQUARE   8 THE CHI SQUARE TEST FOR GOODNESS OF FIT … uses sample data to test hypotheses about the shape or proportions of a population distribution. The test determines how well the obtained sample proportions fit the population proportions specified by the null hypothesis

© aSup-2007 CHI SQUARE   9 THE NULL HYPOTHESIS FOR THE GOODNESS OF FIT  For the chi-square test of goodness of fit, the null hypothesis specifies the proportion (or percentage) of the population in each category  Generally H 0 will fall into one of the following categories: ○ No preference H 0 states that the population is divided equally among the categories ○ No difference from a Known population H 0 states that the proportion for one population are not different from the proportion that are known to exist for another population

© aSup-2007 CHI SQUARE   10 THE DATA FOR THE GOODNESS OF FIT TEST  Select a sample of n individuals and count how many are in each category  The resulting values are called observed frequency (f o )  A sample of n = 40 participants was given a personality questionnaire and classified into one of three personality categories: A, B, or C Category ACategory BCategory C 15196

© aSup-2007 CHI SQUARE   11 EXPECTED FREQUENCIES  The general goal of the chi-square test for goodness of fit is to compare the data (the observed frequencies) with the null hypothesis  The problem is to determine how well the data fit the distribution specified in H 0 – hence name goodness of fit  Suppose, for example, the null hypothesis states that the population is distributed into three categories with the following proportion Category ACategory BCategory C 25%50%25%

© aSup-2007 CHI SQUARE   12 EXPECTED FREQUENCIES  To find the exact frequency expected for each category, multiply the same size (n) by the proportion (or percentage) from the null hypothesis 25% of 40 = 10 individual in category A 50% of 40 = 20 individual in category B 25% of 40 = 10 individual in category C

© aSup-2007 CHI SQUARE   13 THE CHI-SQUARE STATISTIC  The general purpose of any hypothesis test is to determine whether the sample data support or refute a hypothesis about population  In the chi-square test for goodness of fit, the sample expressed as a set of observe frequencies (f o values) and the null hypothesis is used to generate a set of expected frequencies (f e values)

© aSup-2007 CHI SQUARE   14 THE CHI-SQUARE STATISTIC  The chi-square statistic simply measures ho well the data (f o ) fit the hypothesis (f e )  The symbol for the chi-square statistic is χ 2  The formula for the chi-square statistic is χ 2 = ∑ (f o – f e ) 2 fefe

© aSup-2007 CHI SQUARE   15 A researcher has developed three different design for a computer keyboard. A sample of n = 60 participants is obtained, and each individual tests all three keyboard and identifies his or her favorite. The frequency distribution of preference is: Design A = 23, Design B = 12, Design C = 25. Use a chi-square test for goodness of fit with α =.05 to determine whether there are significant preferences among three design LEARNING CHECK

© aSup-2007 CHI SQUARE   16 THE CHI-SQUARE TEST FOR INDEPENDENCE  The chi-square may also be used to test whether there is a relationship between two variables  For example, a group of students could be classified in term of personality (introvert, extrovert) and in terms of color preferences (red, white, green, or blue). REDWHITEGREENBLUE∑ INTRO EXTRO

© aSup-2007 CHI SQUARE   17 OBSERVED AND EXPECTED FREQUENCIES fofo REDWHITEGREENBLUE∑ INTRO EXTRO ∑ REDWHITEGREENBLUE∑ INTRO EXTRO ∑

© aSup-2007 CHI SQUARE   18 THE CHI-SQUARE STATISTIC χ 2 = ∑ (f o – f e ) 2 fefe