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**Ch 15 - Chi-square Nonparametric Methods: Chi-Square Applications**

Chapter 15 Nonparametric Methods: Chi-Square Applications Statistics Is Fun!

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**Nonparametric One-Look.Com Definition:**

Ch 15 - Chi-square Nonparametric One-Look.Com Definition: adjective: not involving an estimation of the parameters of a statistic adjective: not requiring knowledge of underlying distribution: used to describe or relating to statistical methods that do not require assumptions about the form of the underlying distribution You mean we can test without assuming a normal curve? Yes! Statistics Is Fun!

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Ch 15 - Chi-square Goals Conduct a test of hypothesis comparing an observed set of frequencies to an expected set of frequencies Goodness-of-fit tests: Equal Expected Frequencies Unequal Expected Frequencies List the characteristics of the Chi-square distribution We can test a hypothesis with assuming data distribution is normal! Statistics Is Fun!

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**Chi-square (2) Applications**

Ch 15 - Chi-square Chi-square (2) Applications Testing Method where we don’t need assumptions about the shape of the data Testing methods for Nominal data Data with no natural order Examples: Gender Brand preference Color There will be two difference from earlier tests when we do our hypothesis testing: Look up critical value of Chi-square in appendix B Use new formula for Calculated Test Statistic Statistics Is Fun!

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**Goodness-of-fit tests:**

Ch 15 - Chi-square Conduct A Test Of Hypothesis Comparing An Observed Set Of Frequencies To An Expected Set Of Frequencies Goodness-of-fit tests: Equal Expected Frequencies Unequal Expected Frequencies Statistics Is Fun!

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**Purpose Of Goodness-of-fit Tests:**

Ch 15 - Chi-square Purpose Of Goodness-of-fit Tests: Compare an observed distribution (sample) to an expected distribution (population) We will ask the question: Is the difference between the observed values and the expected values: Due to chance (sampling error): The observed distribution is the same as the expected distribution Not due to chance: The observed distribution is not the same as the expected distribution Statistics Is Fun!

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**Hypothesis Testing: Equal Expected Frequencies**

Ch 15 - Chi-square Hypothesis Testing: Equal Expected Frequencies Step 1: State null and alternate hypotheses Ho : There is no significant difference between the set of observed frequencies and the set of expected frequencies H1 : There is a difference between the observed and expected frequencies Step 2: Select a level of significance α = .01 or .05… Statistics Is Fun!

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Ch 15 - Chi-square Hypothesis Testing Step 3: Identify the test statistic (Chi Square = 2) and draw curve with critical value Use α and df to look up critical value in appendix B k = number of categories (k – 1) = degrees of freedom Statistics Is Fun!

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**Hypothesis Testing Step 4: Formulate a decision rule**

Ch 15 - Chi-square Hypothesis Testing Step 4: Formulate a decision rule If our calculated test statistic is greater than , we reject Ho and accept H1, otherwise we fail to reject Ho Statistics Is Fun!

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Ch 15 - Chi-square Hypothesis Testing Equal Expected Frequencies Unequal Expected Frequencies Step 5: Take a random sample, compute the calculated test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses fe will be given or n*% for cell 1st 2nd Statistics Is Fun!

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**Hypothesis Testing Step 5: Conclude: There is either:**

Ch 15 - Chi-square Hypothesis Testing Step 5: Conclude: There is either: The sample evidence suggests that there is not a difference between the observed and expected frequencies The observed distribution is the same as the expected distribution The sample evidence suggests that there is a difference between the observed and expected frequencies The observed distribution is not the same as the expected distribution Statistics Is Fun!

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**List The Characteristics Of The Chi-square Distribution**

Ch 15 - Chi-square List The Characteristics Of The Chi-square Distribution It is positively skewed However, as the degrees of freedom increase, the curve approaches normal It is non-negative Because (fo – fe)2 is never negative There is a family of chi-square distributions df determines which curve to use df = k – 1 k = # of categories Statistics Is Fun!

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**df = 3 df = 5 df = 10 c2 C2 Distribution Ch 15 - Chi-square**

Statistics Is Fun!

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**Limitations Of Chi-Square**

Ch 15 - Chi-square Limitations Of Chi-Square Because fe is used in the denominator, very small fe could result in very large calculated test statistic In General, avoid using Chi-Square when: If there are only two cells: fe >= 5 If there are more than two cells 20% of fe cells contain values less than 5 Statistics Is Fun!

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