14.1 power point

8/7/2019 14.1 Power Point

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Test for GoodnessTest for Goodness

of Fitof FitSection 14.1Section 14.1


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A New TestA New Test

In the previous chapter, we learned how to compareIn the previous chapter, we learned how to comparetwo population proportions. Sometimes, though, wetwo population proportions. Sometimes, though, we

want to examine the distribution of proportions in awant to examine the distribution of proportions in a

single population.single population.

The chiThe chi--square goodnesssquare goodness--ofof--fit (GOF) test allows us tofit (GOF) test allows us todetermine whether a specified population distributiondetermine whether a specified population distribution

seems valid.seems valid.

We can compare two or more population proportionsWe can compare two or more population proportions

using a chiusing a chi--square test for homogeneity.square test for homogeneity.

We can also determine whether the distribution ofWe can also determine whether the distribution of

one variable has been influenced by another variableone variable has been influenced by another variable

using a chiusing a chi--square test of association/independence.square test of association/independence.


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Some ExamplesSome Examples

The methods of this chapter will help usThe methods of this chapter will help us

answer questions such as:answer questions such as:

Are you more likely to have a car accident whenAre you more likely to have a car accident whenusing a cell phone?using a cell phone?

Does background music effect wine purchases?Does background music effect wine purchases?

How does the presence of an exclusive territoryHow does the presence of an exclusive territory

clause in a franchisees contract relate to theclause in a franchisees contract relate to the

success of the business?success of the business?


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The Basic IdeaThe Basic Idea

The idea of the GOF test is to compare theThe idea of the GOF test is to compare the

observed counts for our sample with theobserved counts for our sample with the

counts that we expect from the population.counts that we expect from the population.

The more the observed counts differ fromThe more the observed counts differ from

the expected counts, the more evidence wethe expected counts, the more evidence we

have to reject the null hypothesis.have to reject the null hypothesis.


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Expected CountsExpected Counts

In general, the expected count forIn general, the expected count for

any categorical variable is obtainedany categorical variable is obtained

by multiplying the proportion of theby multiplying the proportion of the

distribution for each category by thedistribution for each category by the

sample size.sample size.


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A New StatisticA New Statistic

The chiThe chi--square statistic (square statistic (XX22)is calculated using)is calculated usingthe formula:the formula:

The larger the difference between the observedThe larger the difference between the observedand expected counts, the largerand expected counts, the largerXX22 will be, andwill be, andthe more evidence there will be againstthe more evidence there will be against HH00..


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The ChiThe Chi--Square DistributionSquare Distribution

The shape of the specific chiThe shape of the specific chi--squaresquare

distribution used to assess the evidencedistribution used to assess the evidence

against Hagainst H00 is determined by the degrees ofis determined by the degrees of

freedom. In a chifreedom. In a chi--square GOF test, thesquare GOF test, the dfdf==(the number of categories(the number of categories 1).1).


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Various ChiVarious Chi--Square DistributionsSquare Distributions


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The ChiThe Chi--Square GOF TestSquare GOF Test

1.1. Name of test:Name of test:22 GOFGOF

2.2. State the Hypotheses:State the Hypotheses:

HH00: the actual population proportions are: the actual population proportions are

equal to the hypothesized proportionsequal to the hypothesized proportions

HHaa: at least one of these proportions is: at least one of these proportions is

incorrectincorrect


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3.3. Check Conditions:Check Conditions:

S.S. SRS from population of interestSRS from population of interest

E.E. All expected counts are at least 5 (or all areAll expected counts are at least 5 (or all are

1 and no more than20% are < 5).1 and no more than20% are < 5).

4.4. Compute theCompute the dfdf, test statistic and the, test statistic and thePP--

valuevalue

5.5. Interpretation in contextInterpretation in context


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Example: A Fair Die?Example: A Fair Die?

Suppose you roll a die 60 times and get 12Suppose you roll a die 60 times and get 12

ones, 9 twos, 10 threes, 6 fours, 11 fives, andones, 9 twos, 10 threes, 6 fours, 11 fives, and

12 sixes. Do you have evidence that the die is12 sixes. Do you have evidence that the die is

unfair?unfair?


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What Did You Expect?What Did You Expect?

60606060TotalTotal

1010121266

101011115510106644

1010101033

101099221010121211

OutcomeOutcomeExpectedExpected

Count, ECount, E

ObservedObserved

Count, OCount, O


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Is It All in the Genes?Is It All in the Genes?

One of the most common applications of the chiOne of the most common applications of the chi--

square GOF test is in the field of genetics. Scientistssquare GOF test is in the field of genetics. Scientists

want to investigate the genetic characteristics ofwant to investigate the genetic characteristics of

offspring that result from mating parents with knownoffspring that result from mating parents with known

genetic makegenetic make--ups. They use rules about dominantups. They use rules about dominant

and recessive genes to predict the ratio of offspringand recessive genes to predict the ratio of offspring

that will fall into each possible genetic category.that will fall into each possible genetic category.

Then the researchers mate the parents and classifyThen the researchers mate the parents and classify

the resulting offspring. The chithe resulting offspring. The chi--square GOF testsquare GOF test

helps the scientists assess the validity of theirhelps the scientists assess the validity of their

hypothesized ratios.hypothesized ratios.


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RedRed--Eyed Fruit FliesEyed Fruit FliesRedRed--Eyed Fruit FliesEyed Fruit Flies

Biologists wish to mate two fruit flies havingBiologists wish to mate two fruit flies having

genetic makegenetic make--up RrSs (R= red eyed, r=up RrSs (R= red eyed, r=

white eyed, S = straightwhite eyed, S = straight--winged, and s =winged, and s =

curlycurly--winged.)winged.)

Use a Punnett square to determine theUse a Punnett square to determine thebiologists predicted ratio.biologists predicted ratio.

Biologists wish to mate two fruit flies havingBiologists wish to mate two fruit flies having

genetic makegenetic make--up RrSs (R= red eyed, r=up RrSs (R= red eyed, r=

white eyed, S = straightwhite eyed, S = straight--winged, and s =winged, and s =

curlycurly--winged.)winged.)

Use a Punnett square to determine theUse a Punnett square to determine thebiologists predicted ratio.biologists predicted ratio.


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Let the Mating Begin!Let the Mating Begin!

To test their hypothesis about theTo test their hypothesis about thedistribution of the offspring, the biologistsdistribution of the offspring, the biologists

mate the fruit flies. Of200 offspring, 99 hadmate the fruit flies. Of200 offspring, 99 hadred eyes and straight wings, 42 had red eyesred eyes and straight wings, 42 had red eyesand curly wings, 49 had white eyes andand curly wings, 49 had white eyes andstraight wings, and 10 had white eyes andstraight wings, and 10 had white eyes and

curly wings. Do these data differcurly wings. Do these data differsignificantly from what the biologistssignificantly from what the biologistspredicted?predicted?

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