How to tell if sample data come from a
Normal distribution
• Plot the data using Histogram – frequency of data values. The
histogram should be bell-shaped.
• Check the empirical rule: For a normal distribution
 68% of the data are within 1 standard deviation of the mean.
 95% of the data are within 2 standard deviation of the mean.
 99.7 of the data are within 3 standard deviation of the mean.
• Another useful plot is called qq-plot (“quantile-quantile” plot). It
plots quantiles of data versus quantile of a normal distribution. If the
data is indeed from a normal distribution we should see a straight
line.
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Likelihood Ratio Tests - Introduction
• Neyman-Pearson lemma provides a method of constructing most
powerful tests for simple hypothesis when the distribution of the
observations is known except for the value of a single unknown
parameter. Sometimes it can be utilized to find uniformly most
powerful test for composite hypothesis that involve a single
parameter.
• In many cases, the distribution of interest has more than one
unknown parameter.
• Likelihood ratio test is a general method used to derive tests of
hypothesis for simple or composite hypotheses.
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Likelihood Ratio Test
• The null hypothesis specifies that the parameter (possibly a vector)
lies in a particular set of possible values denoted by Ω0 and the
alternative hypothesis specifies another set of possible values denoted
by Ωa, which does not overlap with Ω0.
• Examples…
• A likelihood ratio test has a test statistic Λ defined by
 
 
ˆ
L
0

ˆ
L
• For a fixed size α test the decision rule is: reject H0 if Λ ≤ k where k is
determined such that
P(Λ ≤ k | H0) = α.
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Translation of the Likelihood Ratio Test
• Small value of Λ indicates that the likelihood of the sample is
smaller under H0 and therefore the data suggest that H0 is false.
• Large value of Λ indicates no evidence against H0.
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Distribution of the Likelihood Ratio Statistic
• In many cased the distribution of the test statistic Λ is known and
can be used to find k and the rejection region.
• If the distribution of Λ is unknown we use the fact that
 2 ln  ~  (2r )
where r is the number of parameters specified in H0. This result is
true for large n.
• The critical region in this case is: reject H0 if
 2 ln   2, ( r ) .
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Examples
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