2 Nonparametric One-Look.Com Definition: Ch 15 - Chi-squareNonparametricOne-Look.Com Definition:adjective: not involving an estimation of the parameters of a statisticadjective: 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 distributionYou mean we can test without assuming a normal curve?Yes!Statistics Is Fun!
3 Ch 15 - Chi-squareGoalsConduct a test of hypothesis comparing an observed set of frequencies to an expected set of frequenciesGoodness-of-fit tests:Equal Expected FrequenciesUnequal Expected FrequenciesList the characteristics of the Chi-square distributionWe can test ahypothesis withassumingdata distributionis normal!Statistics Is Fun!
4 Chi-square (2) Applications Ch 15 - Chi-squareChi-square (2) ApplicationsTesting Method where we don’t need assumptions about the shape of the dataTesting methods for Nominal dataData with no natural orderExamples:GenderBrand preferenceColorThere will be two difference from earlier tests when we do our hypothesis testing:Look up critical value of Chi-square in appendix BUse new formula for Calculated Test StatisticStatistics Is Fun!
5 Goodness-of-fit tests: Ch 15 - Chi-squareConduct A Test Of Hypothesis Comparing An Observed Set Of Frequencies To An Expected Set Of FrequenciesGoodness-of-fit tests:Equal Expected FrequenciesUnequal Expected FrequenciesStatistics Is Fun!
6 Purpose Of Goodness-of-fit Tests: Ch 15 - Chi-squarePurpose 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 distributionNot due to chance:The observed distribution is not the same as the expected distributionStatistics Is Fun!
7 Hypothesis Testing: Equal Expected Frequencies Ch 15 - Chi-squareHypothesis Testing: Equal Expected FrequenciesStep 1: State null and alternate hypothesesHo : There is no significant difference between the set of observed frequencies and the set of expected frequenciesH1 : There is a difference between the observed and expected frequenciesStep 2: Select a level of significanceα = .01 or .05…Statistics Is Fun!
8 Ch 15 - Chi-squareHypothesis TestingStep 3: Identify the test statistic (Chi Square = 2) and draw curve with critical valueUse α and df to look up critical value in appendix Bk = number of categories(k – 1) = degrees of freedomStatistics Is Fun!
9 Hypothesis Testing Step 4: Formulate a decision rule Ch 15 - Chi-squareHypothesis TestingStep 4: Formulate a decision ruleIf our calculated test statistic is greater than , we reject Ho and accept H1, otherwise we fail to reject HoStatistics Is Fun!
10 Ch 15 - Chi-squareHypothesis TestingEqualExpectedFrequenciesUnequalExpectedFrequenciesStep 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 hypothesesfewill begivenorn*% for cell1st2ndStatistics Is Fun!
11 Hypothesis Testing Step 5: Conclude: There is either: Ch 15 - Chi-squareHypothesis TestingStep 5: Conclude:There is either:The sample evidence suggests that there is not a difference between the observed and expected frequenciesThe observed distribution is the same as the expected distributionThe sample evidence suggests that there is a difference between the observed and expected frequenciesThe observed distribution is not the same as the expected distributionStatistics Is Fun!
12 List The Characteristics Of The Chi-square Distribution Ch 15 - Chi-squareList The Characteristics Of The Chi-square DistributionIt is positively skewedHowever, as the degrees of freedom increase, the curve approaches normalIt is non-negativeBecause (fo – fe)2 is never negativeThere is a family of chi-square distributionsdf determines which curve to usedf = k – 1k = # of categoriesStatistics Is Fun!
13 df = 3 df = 5 df = 10 c2 C2 Distribution Ch 15 - Chi-square Statistics Is Fun!
14 Limitations Of Chi-Square Ch 15 - Chi-squareLimitations Of Chi-SquareBecause fe is used in the denominator, very small fe could result in very large calculated test statisticIn General, avoid using Chi-Square when:If there are only two cells:fe >= 5If there are more than two cells20% of fe cells contain values less than 5Statistics Is Fun!