Goodness of Fit

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• Goodness of Fit
• Multinomials

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Multinomial Proportions

• Thus far we have discussed proportions for
  situations where the result for the qualitative
  variable could be only success or failure
• Now we discuss the situation where there are
  multiple outcomes for the qualitative variable

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EXAMPLE

• Suppose the 1000 people had 5 choices
• COLA OBSERVED
• FREQUENCY
• (1) Coke f1 410
• (2) Pepsi f2 350
• (3) RC f3 80
• (4) Shasta f4 50
• (5) Jolt f5 110

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QUESTIONS

• (1) Can we conclude that there are differences
  in cola preference?
• (2) Last year 40 favored Coke, 35 Pepsi and
  25 all other brands. Can we conclude these
  preferences have changed?
• (3) Give a 95 confidence interval for those who
  favor Coke.

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(1) CAN IT BE CONCLUDED THAT THERE ARE
DIFFERENCES IN COLA PREFERENCES?

• The answer is NO unless we can conclusively show
  otherwise.
• H0 (NO) p1 p2 p3 p4 p5 .2
• HA (YES) At least one pj ? .2
• ? .05
• THIS IS A ?2 (Chi-squared) TEST!

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THE ?2 (Chi-squared) STATISTIC

• The ?2 (Chi-squared) statistic is defined as the
  cumulative mean square differences between the
  observed values (fi) and the expected values if
  H0 were true (ei)

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RULE OF 5

• ?2 (Chi-squared) is actually only an approximate
  distribution for the test statistic
• To be a valid approximation
• ALL eis should be ? 5
• If the rule of 5 is violated, combine some
  categories so that the condition is met

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THE ?2 (Chi-squared) TEST

• Reject H0 if ?2 gt ?2.05,DF
• DF k-1, where k categories (5, here)
• ?2.05,4 a critical value found in a ?2 table
• ?2.05,4 9.48773
• If H0 were true, p1 p2 p3 p4 p5 .2
• We would expect to find e1 .2(1000) 200 and
• e2 .2(1000) 200 e3 .2(1000) 200
  
• e4 .2(1000) 200 e5 .2(1000) 200
• ALL eis ARE ? 5

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CALCULATION OF ?2THE MULTINOMIAL TABLE

• Cola Observed Expected Difference Mean Sq. Dif.
• i fi ei (fi - ei)
  (fi - ei)2/ei
• 1 410 200 210 220.5
• 2 350 200 150 112.5
• 3 80 200 -120 72.0
• 4 50 200 -150 112.5
• 5 110 200 - 90 40.5
• SUM 558.0 ?2

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RESULTS

• ?2 558.0 gt ?2.05,4 9.48773
• There is strong evidence that differences exist
  in cola preferences

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(2) CAN IT BE CONCLUDED COLA PREFERENCES HAVE
CHANGED SINCE LAST YEAR?

• H0 (NO) p1 .40 p2 .35 pOTHER .25
• HA (YES) At least one pj ? its hypothesized
  value
• ? .05
• There are now k 3 categories.
• Reject H0 if ?2 gt ?2.05,2 5.99147

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CALCULATION OF ?2

• Cola Observed Expected Difference Mean Sq. Dif.
• i fi ei (fi - ei)
  (fi - ei)2/ei
• 1 410 400 10 .25
• 2 350 350 0 0
• Other 240 250 -10 .40
• SUM .65 ?2
  
• ?2 .65 lt 5.99147
• Cannot conclude preferences have changed

All eis gt 5
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(3) CONFIDENCE INTERVAL FOR PROPORTION WHO FAVOR
COKE

• This is now binomial
• Coke and everything else

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Excel

• The Excel function CHITEST returns the p-value
  for the hypothesis test.
• Its form is
• CHITEST (Range of observed values,Range of
  estimated values)

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VERY LOW p-value
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Review

• Multinomial problems exist when there are more
  than two possible outcomes for a qualitative
  variable
• Excel Approach -- compare observed values to
  expected values by using CHITEST to give the
  p-value
• Hand approach -- Compare the ? 2 statistic to ? 2
  ? ,DF where DF categories - 1 k-1
• The value of the ? 2 statistic is