Testing Goodness of Fit

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Testing Goodness of Fit
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Definitions / Notation

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Properties of Chi-Square Distribution

• Not symmetric, it is skewed to the right
• Values of the chi-square statistic are always
  greater than or equal to 0 (never negative)

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Finding Chi-Square Statistic

1. Determine degrees of freedom (DF n 1)
2. Determine the area to the right of the desired
   value (may have to divide a into tails in certain
   cases, assume right tail only for this topic)
3. Using DF and Area to the Right, look up the
   chi-square statistic in Table A.4 Critical Values
   for the ?2 Distribution

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Expected Frequencies

• If the probabilities specified by Ho are p1, p2,
  , and the total number of trials is n, the
  expected frequencies are
• E1 np1, E2 np2, and so on

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Hypothesis for Goodness-of-Fit Tests

• The null hypothesis always specifies a
  probability for each category. The alternate
  hypothesis says that some or all of these
  probabilities differ from the true probabilities
  of the categories

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Classical Approach (By Hand)

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Classical Approach (TI-83/84)

• Write down a shortened version of claim
• Come up with null and alternate hypothesis (Ho
  has a probability for each category while H1 says
  that some or all of the actual probabilities
  differ from those specified in Ho)
• See if claim matches Ho or H1
• The picture is always a right tail so put a into
  the right tail
• Place observed frequencies into L1 and expected
  frequencies (np for each category) into L2
• Find critical values (Table A.4 using a and DF
  k 1 where k is the number of categories)
• Find test statistic (?2GOF-Test)
• If test statistic falls in tail, Reject Ho. If
  test statistic falls in main body, Accept Ho.
  Determine the claim based on step 3

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P-Value Approach (TI-84 Plus)

• Write down a shortened version of claim
• Come up with null and alternate hypothesis (Ho
  has a probability for each category while H1 says
  that some or all of the actual probabilities
  differ from those specified in Ho)
• See if claim matches Ho or H1
• Find p-value (X2GOF-Test)
• If p-value is less than a, Reject Ho. If p-value
  is greater than a, Accept Ho. Determine the
  claim based on step 3

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1. Claim

• Following are observed frequencies. The null
  hypothesis is Ho p1 0.25, p2 0.20, p3
  0.50, p4 0.05

1 2 3 4
Category 1 2 3 4
Observed 200 150 350 20
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Check It Out

• In the 1950s-1960s there was a tendency to
  grade on a true curve 10 got As, 20 got
  Bs, 40 got Cs, 20 were Ds, and 10 were Fs.
• Applying this flawed model, an instructor has the
  following distribution 94 As, 19 Bs, 11 Cs, 2
  Ds, and 14 Fs. Does their distribution differ
  from the true curve? Is this instructor
  inflating their grades?