Chapter 11 Other ChiSquared Tests

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Chapter 11 Other Chi-Squared Tests
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Chi-Square Statistic

• Measures how far the observed values are from the
  expected values
• Take sum over all cells in table
• When is large, there is evidence that H0 is
  false.

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Goodness of Fit (GOF) Tests

• We can use a chi-squared test to see if a
  frequency distribution fits a pattern.
• The hypotheses to these tests are written a
  little different than we have seen in the past
  because they are usually written in words.

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GOF Hypothesis Test

• A researcher wishes to see of the number of
  adults who do not have health insurance is
  equally distributed among three categories

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GOF Hypothesis Test

• Step 1
• Ho The number of people who do not have health
  insurance is equally distributed over the three
  categories.
• Ha The number of people who do not have health
  insurance is not equally distributed over the
  three categories.

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GOF Hypothesis Test

• Step 2
• a 0.05
• Step 3

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GOF Hypothesis Test

• Step 4
• Test Statistic 8.1
• Put Observed in L1 and Expected in L2
• L3 (L1-L2)2 / L2
• Sum L3
• ?2 cdf (test stat, E99, df) p-value 0.017

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GOF Hypothesis Test

• Step 5
• Reject Ho
• Step 6
• There is enough evidence to suggest that the
  number of people who do not have health insurance
  is not equally distributed over the three
  categories.

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Two-Way Tables

• Summarizes the relationship between two
  categorical variables
• Row values for one categorical variable
• Column values for other categorical variable
• Table entries number in row by column class

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Example of Two-Way Table
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Independence of Categorical Variables.

• Is there a relationship between two categorical
  variables. Are two variables related to each
  other?
• Unrelated independent
• Related dependent

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Example

• A 1992 poll conducted by the University of
  Montana classified respondents by sex and
  political party.
• Sex Male and Female
• Party Democrat, Republican, Independent
• Is there evidence of an association between
  gender and party affiliation?

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Hypothesis Test for Independence

• HO Sex and political party affiliation are
  independent (have no relationship).
• HA Sex and political party affiliation are
  dependent (are related to each other.)

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Two-Way Tables

• Describe table with of rows (r) and of
  columns (c)
• r x c table
• Each number in table is called a cell
• r times c cells in table
• We will use the two-way table to test our
  hypotheses.

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Data
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Expected Counts

• If HO is true, we would expect to get a certain
  number of counts in each cell.
• Expected cell count row total column total
• table total

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Example of Expected Counts

• Cell Male and Democrat
• Expected Count
• Cell Male and Republican
• Expected Count
• Cell Male and Independent
• Expected Count

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Two-Way Table of Expected Counts
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Expected Counts

• Expected cell count is close to observed cell
  count
• Evidence Ho is true
• Expected cell count is far from observed cell
  count
• Evidence Ho is false

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Chi-Squared Test for Independence

• To test these hypotheses, we will use a
  Chi-Squared Test for Independence if the
  assumptions hold.
• Assumptions
• Expected Cell Counts are all gt 5

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Chi-Square Statistic

• Measures how far the observed values are from the
  expected values
• Take sum over all cells in table
• When is large, there is evidence that H0 is
  false.
• Your calculator will do this for you.

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Chi-Square Statistic

• Cell Male and Democrat
• Observed Count 36
• Expected Count 43.66
• Cell Male and Republican
• Observed Count 45
• Expected Count 40.54

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Chi-Square Statistic

• Repeat this process for all 6 cells
• ?2 4.85
• As long as the assumptions are met
• ?2 will have a ?2 distribution with d.f.
  (r-1)(c-1)
• Sex has 2 categories (so r 2) party has 3
  categories (so c 3)
• We have (2-1)(3-1) 2 degrees of freedom

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Hypothesis Test

• P-value P(?2 gt 4.85) 0.09
• Decision Since p-value gt a 0.05, we will Do
  Not Reject HO.
• Conclusion There is no evidence of a
  relationship between sex and political party
  affiliation.

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Homogeneity of Proportions

• Samples are selected from different populations
  and a researcher wants to determine whether the
  proportion of elements are common for each
  population.
• The hypotheses are
• Ho p1 p2 p3 p4
• Ha At least one is different

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Homogeneity of Proportions

• An advertising firm has decided to ask 92
  customers at each of three local shopping malls
  if they are willing to take part in a market
  research survey. According to previous studies,
  38 of Americans refuse to take part in such
  surveys. At a 0.01, test the claim that the
  proportions are equal.

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Homogeneity of Proportions

• Step 1
• Ho p1 p2 p3
• Ha At least one
• is different
• Step 2
• a 0.01
• Step 3

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Homogeneity of Proportions

• Step 4
• Put into your calculator
• Observed in matrix A
• Expected in matrix B
• Test statistic 5.602
• P-value 0.06

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Homogeneity of Proportions

• Step 5
• Do Not Reject Ho
• Step 6
• There is not sufficient evidence to suggest that
  at least one is different.