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AP STATISTICS LESSON 13 – 1 (DAY 1) CHI-SQUARE PROCEDURES TEST FOR GOODNESS OF FIT ESSENTIAL QUESTION: 2 What is X , goodness of fit, and how are they used in statistics? Objectives: • To use data to find X2 • To use X2 to find the probability of a sample fitting a population. Introduction Sometimes we want to examine the distribution of proportions in a single population. The chi-square test for goodness of fit allows us to determine whether a specified population distribution seems valid. We can compare two or more population proportions using chi-square test for homogeneity of populations. Introduction (continued…) In doing so, we will organize our data in a two-way table. It is also possible to use the information provided in a twoway table to determine whether the distribution of one variable has been influenced by another variable. The chi-square test of association/independence helps us decide this issue. Tests for Goodness of Fit There is a single test that can be applied to see if the observed sample distribution is significantly different from the hypothesized population distribution. It is called the chi-square (χ2) test for goodness of fit. Before proceeding with a significance test, it’s always a good idea to plot the data. Χ2 Formula We need way to determine how well the observed counts ( O ) fit the expected counts ( E ) under Ho. The procedure is to calculate the quantity ( O – E )2 E for each category and then add up these terms. The sum is labeled Χ2 and is called the chi-square statistic. X 2 Formula (continued…) Degrees of freedom is determined by taking the number of categories and subtracting 1. The chi-square test statistic is a point on the horizontal axis, and the area to the right is the Pvalue of the test. This P-value is the probability of observing a value X 2 at least as extreme as the one actually observed. Use Table E (Chi-squared Distribution Critical Values) to find the P-values. The CHI-SQUARE DISTRIBUTIONS The chi-square distributions are a family of distributions that take only positive values and are skewed to the right. A specific chi-square distribution is specified by one parameter, called the degrees of freedom. The Chi-square Density Curve Properties 1. 2. The total area under a chi-square curve is equal to 1. Each chi-square curve (except when df = 1) begins at 0 on the horizontal axis, increases to a peak, and then approaches the horizontal axis asymptotically from above. 3. Each chi-square curve is skewed to the right. As the number of degrees of freedom increases, the curve becomes more and more symmetrical and looks more like a normal curve. Example 13.1 Page 728 The Graying of America With better medicine and healthier lifestyles, people are living longer. Consequently, a larger percentage of the population is of retirement age. We want to determine if the distribution of age groups in the US in 1996 as changed significantly from the 1980 distribution. Test the following Hypothesis: H0: age group dist. In 1996 is the same as the 1980 dist. Ha: age group dist. In 1996 is the different from as the 1980 dist. Goodness of Fit Test The chi-square test for goodness of fit can be applied to see if the observed sample distribution is significantly different from the hypothesized population distribution. A goodness of fit test is used to help determine whether a population has a certain hypothesized distribution, expressed as proportions of population members falling into various outcome categories. Goodness of Fit (continued…) To test the hypothesis • Ho : the actual population proportions are equal to hypothesized proportions – First calculate the chi-square test statistic X2 = ∑ ( O – E )2 / E – Then X2 distribution with ( n – 1 ) degrees of freedom. • For a test of Ho against the alternative hypothesis – Ha : the actual population proportion are different from the hypothesized proportions the P-value is P( x2 ≥ X2 ). Goodness of Fit: Conditions Conditions: You may use this test with critical values from the chi-square distribution when all individual expected counts are at least 1 and no more than 20% of the expected counts are less than 5. Example 13.2 Page 733 Red-eyed Fruit Flies The most common application of the chisquare goodness of fit test is in the field of genetics. In this example, it is used to investigate the genetic characteristics of offspring that result from mating parents with known genetic makeups.