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## The CHI SQUARE Statistic

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**The CHI SQUARE Statistic**Tests for Goodness of Fit and Independence**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)**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**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**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**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**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**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**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 H0 will fall into one of the following categories: • No preference H0 states that the population is divided equally among the categories • No difference from a Known population H0 states that the proportion for one population are not different from the proportion that are known to exist for another population**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 (fo) • A sample of n = 40 participants was given a personality questionnaire and classified into one of three personality categories: A, B, or C**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 H0 – hence name goodness of fit • Suppose, for example, the null hypothesis states that the population is distributed into three categories with the following proportion**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**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 (fovalues) and the null hypothesis is used to generate a set of expected frequencies (fe values)**THE CHI-SQUARE STATISTIC**• The chi-square statistic simply measures ho well the data (fo) fit the hypothesis (fe) • The symbol for the chi-square statistic is χ2 • The formula for the chi-square statistic is (fo – fe)2 χ2 = ∑ fe**LEARNING CHECK**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**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).**THE CHI-SQUARE STATISTIC**(fo – fe)2 χ2 = ∑ fe