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					Statistics

HYPOTHESIS TESTING
TERMS
 Null hypothesis– The claim being assessed in a
  hypothesis test is called the null hypothesis.
 Usually, the null hypothesis is a statement of
  “no change from the traditional value,” “no
  effect,” “no difference,” or “no relationship.”
 For a claim to be a testable null hypothesis, it
  must specify a value for some population
  parameter that can form the basis for
  assuming a sampling distribution for a test
  statistic.
TERMS

 Alternative hypothesis—The alternative
  hypothesis proposes what we should conclude
  if we find the null hypothesis to be unlikely.
 Two-sided alternative–An alternative hypothesis
  is two-sided HA : p  p0  when we are interested
  in deviations in either direction away from the
  hypothesized parameter value.

        
TERMS

   One-sided alternative– An alternative
    hypothesis is one-sided (e.g., HA: p > p0 or
    HA: p < p0) when we are interested in deviations
    in only one direction away from the
    hypothesized parameter value.
TERMS

 P-value– The probability of observing a value for a
  test statistic at least as far from the hypothesized
  value as the statistic value actually observed if the
  null hypothesis is true.
 A small P-value indicates either that the
  observation is improbable or that the probability
  calculation was based on incorrect assumptions.
 The assumed truth of the null hypothesis is the
  assumption under suspicion.
TERMS

   One-proportion z-test– A test of the null
    hypothesis that the proportion of a single
    sample equals a specified value (H0: p = p0) by
                                p  p0
                                ˆ
    referring the statistic z         to a Standard
                                SD p
                                    ˆ
    Normal model.


                
HYPOTHESIS TESTING

 A hypothesis proposes a model for the world.
 Check the data

 Is the data consistent with the model?
WHAT IS A HYPOTHESIS?

 A hypothesis is like a jury trial.
 The jury starts by assuming that the person is
  innocent.
 The jury then needs to prove the person guilt
  beyond a reasonable doubt.
 Then and only then can the jury reject the
  hypothesis of innocence and declare the
  person guilty.
HYPOTHESIS TESTING

 Statistics is similar except that we quantify the
  level of doubt.
 If the data is surprising, but we believe it is
  trustworthy, then we doubt our hypothesis.
HYPOTHESIS TESTING

 What do we mean by surprising?
 An event that has a low probability of occurring
  is a surprise by definition.
 We can look at the probability that the event
  could have happened by chance.
 The probability quantifies how surprised we
  are.
 This is the P-value.
HYPOTHESIS TESTING

   Start by assuming the hypothesis is true.
 The null hypothesis denoted by H0, specifies
  the population model parameter of interest and
  proposes a value for that parameter.
 Written: H0: parameter = hypothesized value.

 The values comes from the Who and What of
  the data.
HYPOTHESIS TESTING– COURTROOM EXAMPLE

 We are still in the jury trial.
 Suppose the defendant is accused of robbery.
 The data is gathered when the lawyers present
  evidence either for or against the defendant.
 The jury has to use “hypothesis testing” to
  determine if the defendant is guilty beyond a
  reasonable doubt.
 The jury has to determine the degree to which
  the evidence contradicts the presumption of
  innocence.
GUILTY VS. INNOCENT

 A jury is given the duty to decide the level of
  innocence if any.
 Upon looking at the evidence, they may decide
  that the person is “not guilty.”
 They did not say he was “innocent.”

 Not guilty means there was not enough
  evidence to prove him guilty.
 So we “failed to reject” the hypothesis.
REASONING OF HYPOTHESIS TESTING

 Hypothesis tests follow a carefully structured
  path.
 There are four sections that need to be dealt
  with.
 Hypothesis

 Model

 Mechanics

 Conclusion
REASONING OF HYPOTHESIS TESTING
 Hypotheses—
 State the null hypothesis—
     Translatethe question into a statement with
      parameters so that it can be tested.
     H0: parameter = hypothesized value

   State the alternative hypothesis
     Contains  the values of the parameter we accept if
      we reject the null hypothesis
     HA represents the alternative hypothesis
REASONING OF HYPOTHESIS TESTING
 Model—
 To plan a statistical hypothesis test, specify the
  model you will use to test the null hypothesis and
  the parameter of interest.
 Models require an assumption so you will need to
  state them and check any corresponding
  conditions.
 The model should end with a statement like:
  Because the conditions are satisfied, I can model
  the sampling distribution with a Normal model.
REASONING OF HYPOTHESIS TESTING

 Models—
 This is the place you run your tests

 There are many tests that can be run:
     One-proportion z-test
     One-proportion t-test

     Two-proportion z-test

     Two-proportion t-test

     Chi-squared  
                    2 test
REASONING OF HYPOTHESIS TESTING

 The test about proportions is called a one-
  proportion z-test.
 The conditions for the one-proportion z-test are
  the same for the one-proportion z-interval.
 We test the hypothesis H0: p = p0 using the
  statistic    p  p0 
                ˆ
           z
                SD p
                    ˆ



    
REASONING OF HYPOTHESIS TESTING

   We use the hypothesized proportion to find the
    standard deviation,        p0q0
                        SD p 
                          ˆ            .
                                   n

   When the conditions are met and the null
    hypothesis is true, the static follows the Normal
               
    model, so we can use the model to obtain a P-
    value.
REASONING OF HYPOTHESIS TESTING
 Mechanics—
 This is the section we use our calculations of
  our test statistic from the data.
 Different tests require different formulas.
 The ultimate calculation is to obtain a P-value–
  the probability that the observed statistic value
  (or an even more extreme value) could occur if
  the null model were correct.
 If the P-value is small enough, we will reject the
  null hypothesis.
REASONING OF HYPOTHESIS TESTING

 Conclusion—
 This is a statement about the null hypothesis.

 The conclusion must state either that we reject
  or that we fail to reject the null hypothesis.
 The conclusion is always stated in context.
ALTERNATIVES

 An alternative hypothesis is known as a two-
  sided alternative.
 We are equally interested in deviations on
  either side of the null hypothesis value.
 For two-sided alternatives, the P-value is the
  probability of deviating on either direction from
  the null hypothesis value.
ALTERNATIVES

 How do you determine if you need a two-sided
  test?
 Look at the W’s specifically the Why of the
  study.
 The way the null hypothesis is stated is will
  determine if you are doing a one-sided or two-
  sided test.
WHAT CAN GO WRONG?

 Don’t base your null hypothesis on what you
  see in the data.
 Don’t base your alternative hypothesis on the
  data, either.
 Don’t make your null hypothesis what you want
  to show to be true.
 Don’t forget to check your conditions.
WHAT WE HAVE LEARNED
 We start with a null hypothesis specifying the
  parameter of a model we will test using our
  data.
 Our alternative hypothesis can be one- or two-
  sided, depending on what we want to learn.
 We must check the appropriate assumptions
  and conditions before proceeding with our test.
 If the data are out of line with the null
  hypothesis model, the P-value will be small and
  we will reject the null hypothesis.
WHAT WE HAVE LEARNED

 If the data are consistent with the null
  hypothesis model, the P-value will be small and
  we will reject the null hypothesis.
 We must always state our conclusion in the
  context of the original question.

				
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posted:6/14/2013
language:English
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