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### Chi square

• 1. © aSup-2007 CHI SQUARE    1 The CHI SQUARE Statistic Tests for Goodness of Fit and Independence
• 2. © 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)
• 3. © 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
• 4. © 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
• 5. © 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
• 6. © 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
• 7. © 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
• 8. © 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
• 9. © 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 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
• 10. © 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 (fo)  A sample of n = 40 participants was given a personality questionnaire and classified into one of three personality categories: A, B, or C Category A Category B Category C 15 19 6
• 11. © 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 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 Category A Category B Category C 25% 50% 25%
• 12. © 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
• 13. © 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 (fovalues) and the null hypothesis is used to generate a set of expected frequencies (fe values)
• 14. © aSup-2007 CHI SQUARE    14 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 χ2 = ∑ (fo – fe)2 fe
• 15. © 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
• 16. © aSup-2007 CHI SQUARE    16 Dari https://twitter.com/#!/palangmerah diketahui bahwa persentase golongan darah di Indonesia adalah: A : 25,48%, B : 26,68%, O : 40,77 %, AB : 6,6 % Golongan darah di kelas kita? Apakah berbeda dengan data PMI? LEARNING CHECK
• 17. © aSup-2007 CHI SQUARE    17 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). RED WHITE GREEN BLUE ∑ INTRO 10 3 15 22 50 EXTRO 90 17 25 18 150 100 20 40 40 200
• 18. © aSup-2007 CHI SQUARE    18 OBSERVED AND EXPECTED FREQUENCIES fo RED WHITE GREEN BLUE ∑ INTRO 10 3 15 22 50 EXTRO 90 17 25 18 150 ∑ 100 20 40 40 200 fe RED WHITE GREEN BLUE ∑ INTRO 50 EXTRO 150 ∑ 100 20 40 40 200
• 19. © aSup-2007 CHI SQUARE    19 OBSERVED AND EXPECTED FREQUENCIES fo RED WHITE GREEN BLUE ∑ INTRO 10 3 15 22 50 EXTRO 90 17 25 18 150 ∑ 100 20 40 40 200 fe RED WHITE GREEN BLUE ∑ INTRO 25 5 10 10 50 EXTRO 75 15 30 30 150 ∑ 100 20 40 40 200
• 20. © aSup-2007 CHI SQUARE    20 OBSERVED AND EXPECTED FREQUENCIES fo R W G B ∑ INTRO 10 3 15 22 50 EXTRO 90 17 25 18 150 ∑ 100 20 40 40 200 (fo– fe)2 R W G B INTRO EXTRO fe R W G B ∑ INTRO 25 5 10 10 50 EXTRO 75 15 30 30 150 ∑ 100 20 40 40 200
• 21. © aSup-2007 CHI SQUARE    21 OBSERVED AND EXPECTED FREQUENCIES fo R W G B ∑ INTRO 10 3 15 22 50 EXTRO 90 17 25 18 150 ∑ 100 20 40 40 200 (fo– fe)2 R W G B INTRO (-15)2 (-2)2 (5)2 (12)2 EXTRO (15)2 (-2)2 (-5)2 (-12)2 fe R W G B ∑ INTRO 25 5 10 10 50 EXTRO 75 15 30 30 150 ∑ 100 20 40 40 200
• 22. © aSup-2007 CHI SQUARE    22 OBSERVED AND EXPECTED FREQUENCIES (fo– fe)2 /fe R W G B INTRO EXTRO fe R W G B INTRO 25 5 10 10 EXTRO 75 15 30 30 (fo– fe)2 R W G B INTRO 225 4 25 144 EXTRO 225 4 25 144
• 23. © aSup-2007 CHI SQUARE    23 OBSERVED AND EXPECTED FREQUENCIES (fo– fe)2 /fe R W G B INTRO 9 0,8 2,5 14,4 EXTRO 3 0,267 0,833 4,8 fe R W G B INTRO 25 5 10 10 EXTRO 75 15 30 30 (fo– fe)2 R W G B INTRO 225 4 25 144 EXTRO 225 4 25 144
• 24. © aSup-2007 CHI SQUARE    24 THE CHI-SQUARE STATISTIC χ2 = ∑ (fo – fe)2 fe χ2 = 35,6 df = (C-1) (R-1) = (3) (1) = 3 χ2 critical at α = .05 is 7,81
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