Inferences for Categorical Data

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Inferences for Categorical Data

• Lecture 1

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Comparing Two Proportions

• Section 9.4

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Example 1.

• Mailing letters. Summarize data by giving sample
  proportion of all letters that would be mailed.
• Construct a 95 confidence interval for the
  proportion of all Hope students that would mail
  such a letter (consider the people in this class
  to be representative of the population).

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Proportions - Large Sample
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Notation

• p1 The proportion of successes in the first
  population.
• p2 The proportion of successes in the second
  population.
• The proportion of successes in the sample
  from the first population.
• The proportion of successes in the sample
  from the second population.
• The proportion of successes in the combined
  samples.

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Example 2.

• Summarize data.
• Illustrate Data.

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Example 2, continued.

• Is there significant evidence at the a 0.05
  level that ________________________.
• Give the hypotheses, test statistic, P-value, and
  conclusion.

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Example 2, continued.

• Construct a 95 confidence interval for p1-p2.

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Assumptions

• The data consist of two independent SRSs from
  both populations of interest.
• Both populations are at least 10 times as large
  as their respective sample so that sampling with
  replacement and sampling without replacement are
  essentially the same.
• The sample sizes are large. A rule of thumb
  given in some texts is that the counts of
  successes and failures in both samples are at
  least 5.
• Check if the assumptions hold for Example 2.

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Where we are going,Chapter 14

• Binomial Data. A population consists of only two
  outcomes successes and failures or 0s and 1s.
  Get a sample of given size from this population
  and record the number of successes.
• Multinomial Data. The population is divided into
  k??2 categories. Get a sample of given size
  from this population and record the number in
  each category.
• We have studied one sample and two sample
  inferences for proportions. These are both
  applicable to binomial data. Chapter 14 extends
  these ideas to Multinomial data.
• Notation.
• 
  

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Outline
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Goodness of Fit Tests

• Section 14.1
• (Section 14.2 in lab tomorrow.)

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Example

• M and Ms advertises that plain m and ms are 30
  brown, 20 yellow and red, and 10 orange, blue,
  and green. Does your bag of m and ms show
  significant deviation from this?
• Give null and alternative hypotheses.
• Note that alternative is Many Sided.

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Example, Computations
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The test statistic

• The applicable test statistic is the sum
  of(O-E)2/E over all categories. This test
  statistic is called c2.
• The larger the test statistic is, the more
  significant the deviation from the null
  hypothesis is. Thus, the critical region will be
  of form c2 ? c for some c.
• Theorem. Provided that the count in each
  category is at least 5, assuming the null
  hypothesis, is true the test statistic has an
  approximate c2 distribution with k-1 degrees of
  freedom. (k number of categories.)
• Critical values for c2 are in table A.7 (pg 727).

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Conclusion of Example

• What is the 5 critical region for the m
  and ms test?
• Draw the appropriate conclusion.
• Table A.11 can be used to more accurately
  estimate P-value.

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Summary

• Section 8.3 Tests for binomial data when a null
  value is specified. (Small or large samples.)
• Section 9.4 A comparative test for binomial
  data. (Only large sample test.)
• In the above two sections we learned procedures
  for confidence intervals and hypothesis tests.
  The alternative hypothesis can be one or two
  sided.
• Chapter 14
• Section 1 and 2 An extension of 8.3 to
  multinomial data. The m and ms example is a
  template for 14.1 problems.
• Section 3 An extension of 9.4 to multinomial
  data.
• Only study hypotheses tests. There is only one
  possible alternative and it is many sided.