Lecture27

Lecture27 - STAT 350 Lecture 27 8.3 Tests Concerning...

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4/27/2011 1 STAT 350 Lecture 27 8.3 Tests Concerning Hypotheses About a Categorical Population 8.3 Tests Concerning Hypotheses About a Categorical Population Examples of Categorical Data Gender Color or Type of Transmissions of automobiles Political Party Registration Chi-square Test Invented in 1900 by Karl Pearson Example 1 A gambler is accused of using a loaded die, but he pleads innocent. A record has kept of the last 60 throws. There is a disagreement about how to interpret the data and a statistician is called in. value 1 2 3 4 5 6 sum Observed 4 6 17 16 8 9 60 expected Example 1 A gambler is accused of using a loaded die, but he pleads innocent. A record has kept of the last 60 throws. There is a disagreement about how to interpret the data and a statistician is called in. value 1 2 3 4 5 6 sum Observed 4 6 17 16 8 9 60 expected 10 10 10 10 10 10 60 Chi-square test idea: combine all these differences into one overall measure of the distance between observed and expected value. χ 2 = ∑(observed – expected ) 2 /expected χ 2 has chi-square distribution with degree of freedom = #categories -1 Example 1 χ 2 = ∑(observed – expected ) 2 /expected χ 2 = 6 2 /10 + 4 2 /10 + 7 2 /10 + 6 2 /10 + 2 2 /10+ 1 2 /10 = 14.2 df = 5 value 1 2 3 4 5 6 sum Observed 4 6 17 16 8 9 60 expected 10 10 10 10 10 10 60
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4/27/2011 2 Example 1 Hypothesis: H0: π1=π2=π3=π4=π5=π6=1/6 Ha: die is not fair (H0 is incorrect) Test Statistic: χ 2 = 14.2, with df=5 P-value < 0.015 => reject H0 at α =0.05 Table VII (page 571): Right tail area = 0.015 when χ 2 = 14.09 Chi-Squared Distributions A family of distributions, each member is identified by the degree of freedom
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Lecture27 - STAT 350 Lecture 27 8.3 Tests Concerning...

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