ChiSquare Tests and the FDistribution

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Chapter 10

• Chi-Square Tests and the F-Distribution

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Chapter Outline

• 10.1 Goodness of Fit
• 10.2 Independence
• 10.3 Comparing Two Variances
• 10.4 Analysis of Variance

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Section 10.1

• Goodness of Fit

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Section 10.1 Objectives

• Use the chi-square distribution to test whether a
  frequency distribution fits a claimed distribution

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Multinomial Experiments

• Multinomial experiment
• A probability experiment consisting of a fixed
  number of trials in which there are more than two
  possible outcomes for each independent trial.
  
• A binomial experiment had only two possible
  outcomes.
  
• The probability for each outcome is fixed and
  each outcome is classified into categories.
  

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Multinomial Experiments

• Example
• A radio station claims that the distribution of
  music preferences for listeners in the broadcast
  region is as shown below.
  

Each outcome is classified into categories.
The probability for each possible outcome is
fixed.
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Chi-Square Goodness-of-Fit Test

• Chi-Square Goodness-of-Fit Test
• Used to test whether a frequency distribution
  fits an expected distribution.
  
• The null hypothesis states that the frequency
  distribution fits the specified distribution.
  
• The alternative hypothesis states that the
  frequency distribution does not fit the specified
  distribution.
  

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Chi-Square Goodness-of-Fit Test

• Example
• To test the radio stations claim, the executive
  can perform a chi-square goodness-of-fit test
  using the following hypotheses.
  

H0 The distribution of music preferences in the
broadcast region is 4 classical, 36 country,
11 gospel, 2 oldies, 18 pop, and 29 rock.
(claim) Ha The distribution of music preferences
differs from the claimed or expected
distribution.
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Chi-Square Goodness-of-Fit Test

• To calculate the test statistic for the
  chi-square goodness-of-fit test, the observed
  frequencies and the expected frequencies are
  used.
• The observed frequency O of a category is the
  frequency for the category observed in the sample
  data.
  

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Chi-Square Goodness-of-Fit Test

• The expected frequency E of a category is the
  calculated frequency for the category.
  
• Expected frequencies are obtained assuming the
  specified (or hypothesized) distribution. The
  expected frequency for the ith category is Ei
  npi
• where n is the number of trials (the sample
  size) and pi is the assumed probability of the
  ith category.

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Example Finding Observed and Expected Frequencies

• A marketing executive randomly selects 500 radio
  music listeners from the broadcast region and
  asks each whether he or she prefers classical,
  country, gospel, oldies, pop, or rock music. The
  results are shown at the right. Find the observed
  frequencies and the expected frequencies for each
  type of music.

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Solution Finding Observed and Expected
Frequencies

• Observed frequency The number of radio music
  listeners naming a particular type of music

observed frequency
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Solution Finding Observed and Expected
Frequencies

• Expected Frequency Ei npi

500(0.04) 20
500(0.36) 180
500(0.11) 55
500(0.02) 10
500(0.18) 90
500(0.29) 145
n 500
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Chi-Square Goodness-of-Fit Test

• For the chi-square goodness-of-fit test to be
  used, the following must be true.
  
• The observed frequencies must be obtained by
  using a random sample.
  
• Each expected frequency must be greater than or
  equal to 5.
  

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Chi-Square Goodness-of-Fit Test

• If these conditions are satisfied, then the
  sampling distribution for the goodness-of-fit
  test is approximated by a chi-square distribution
  with k 1 degrees of freedom, where k is the
  number of categories.
• The test statistic for the chi-square
  goodness-of-fit test is
  
• where O represents the observed frequency of
  each category and E represents the expected
  frequency of each category.
  

The test is always a right-tailed test.
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Chi-Square Goodness-of-Fit Test
In Words In Symbols

• Identify the claim. State the null and
  alternative hypotheses.
  
• Specify the level of significance.
• Identify the degrees of freedom.
• Determine the critical value.

State H0 and Ha.
Identify ?.
d.f. k 1
Use Table 6 in Appendix B.
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Chi-Square Goodness-of-Fit Test
In Words In Symbols

• Determine the rejection region.
• Calculate the test statistic.
• Make a decision to reject or fail to reject the
  null hypothesis.
  
• Interpret the decision in the context of the
  original claim.

If ?2 is in the rejection region, reject H0.
Otherwise, fail to reject H0.
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Example Performing a Goodness of Fit Test

• Use the music preference data to perform a
  chi-square goodness-of-fit test to test whether
  the distributions are different. Use a 0.01.
  

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Solution Performing a Goodness of Fit Test

• H0
• Ha
• a
• d.f.
• Rejection Region

music preference is 4 classical, 36 country,
11 gospel, 2 oldies, 18 pop, and 29 rock
music preference differs from the claimed or
expected distribution

• Test Statistic
• Decision
• Conclusion

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Solution Performing a Goodness of Fit Test
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Solution Performing a Goodness of Fit Test

• H0
• Ha
• a
• d.f.
• Rejection Region

music preference is 4 classical, 36 country,
11 gospel, 2 oldies, 18 pop, and 29 rock
music preference differs from the claimed or
expected distribution

• Test Statistic
• Decision

?2 22.713
Reject H0
There is enough evidence to conclude that the
distribution of music preferences differs from
the claimed distribution.
22.713
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Example Performing a Goodness of Fit Test

• The manufacturer of MMs candies claims that the
  number of different-colored candies in bags of
  dark chocolate MMs is uniformly distributed. To
  test this claim, you randomly select a bag that
  contains 500 dark chocolate MMs. The results
  are shown in the table on the next slide. Using a
  0.10, perform a chi-square goodness-of-fit test
  to test the claimed or expected distribution.
  What can you conclude? (Adapted from Mars
  Incorporated)

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Example Performing a Goodness of Fit Test

• Solution
• The claim is that the distribution is uniform, so
  the expected frequencies of the colors are equal.
  
  
• To find each expected frequency, divide the
  sample size by the number of colors.
  
• E 500/6 83.3

n 500
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Solution Performing a Goodness of Fit Test

• H0
• Ha
• a
• d.f.
• Rejection Region

Distribution of different-colored candies in bags
of dark chocolate MMs is uniform
Distribution of different-colored candies in bags
of dark chocolate MMs is not uniform

• Test Statistic
• Decision
• Conclusion

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Solution Performing a Goodness of Fit Test
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Solution Performing a Goodness of Fit Test

• H0
• Ha
• a
• d.f.
• Rejection Region

Distribution of different-colored candies in bags
of dark chocolate MMs is uniform
Distribution of different-colored candies in bags
of dark chocolate MMs is not uniform

• Test Statistic
• Decision

?2 3.016
Fail to Reject H0
There is not enough evidence to dispute the claim
that the distribution is uniform.
3.016
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Section 10.1 Summary

• Used the chi-square distribution to test whether
  a frequency distribution fits a claimed
  distribution

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