Hypothesis Testing Introduction by ubb16013

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									Hypothesis Testing:
   Introduction
     Part 1 of 5
      Lecture 27
     Section 11.1

                      1
        ECO220Y: Overview
First Steps:     Theory:         Inference:

  Form a
 Research       Probability
 Question                        Estimation
               Distribution of
   Data         a Random
 Collection       Variable
                                 Hypothesis
 Describe                         Testing
   the          Sampling
 Sample        Distributions

                                              2
  Hypothesis Testing Overview
• Make hypotheses about the unknown
  population parameters
  – Ex: Hypothesize that population mean is 20
• Collect data and compute sample statistics
• Conduct statistical tests of hypotheses
  – Use sample statistics, sample size, sampling
    distributions
  – What does evidence say about hypotheses?

                                                   3
 Analogy: Parameters and Trial
• Population parameter     • Guilt of the defendant
  is unknown                 is unknown
• Make hypotheses          • Make hypotheses on
  about the parameter        defendant’s guilt
• Collect a sample from    • Collect evidence
  the population and         relevant to the
  calculate statistics       determination of guilt
  – Observe entire pop.?     – Complete evidence?
• Test hypotheses          • Test hypotheses

                                                      4
           Two Hypotheses
• Null hypothesis (H0): Initial presumption
  about the unknown parameter, which is
  not based on evidence (data)
• Alternative or Research Hypothesis (H1):
  Can be proved based on evidence (data)
• Example:
  – H0: Defendant is innocent
  – H1: Defendant is guilty

                                              5
     Jury Makes an Inference
• H0: Person is innocent; H1: Person is guilty
• Acquit: insufficient evidence to infer guilt
  – Fail to reject presumption of innocence (H0)
     • Does this mean there is no evidence of guilt?
     • If you are acquitted does that prove innocence?
     • Does failing to reject H0 mean it is true?
• Convict: enough evidence to infer guilt (H1)
  – Reject presumption of innocence (H0)
     • Jury convinced of guilt beyond a reasonable doubt
                                                           6
       Setting Up Hypotheses
• Usual legal standard:       • If legal standard:
  “Innocent until proven        “Guilty until proven
  guilty beyond a               innocent beyond a
  reasonable doubt”             reasonable doubt”
   – H0:                         – H0:
   – H1:                         – H1:
  – If no evidence, what         – If no evidence, what
    must the jury do:              must the jury do:
    acquit or convict?             acquit or convict?
    H0 is initial presumption about the population:
            it is not based on any evidence               7
     Take Notice of Language
• In statistics, like in a trial by jury, we either:
  – Reject the null hypothesis in favor of the
    research hypothesis
  – Or fail to find enough evidence to reject the
    null hypothesis (find insufficient evidence to
    support the research hypothesis)
• What we cannot do with a sample:

                                   X
  – “Prove” the null hypothesis
  – “Accept” the null hypothesis
                                                     8
                R. A. Fisher
“In relation to any experiment we may speak of this
hypothesis as the ‘null hypothesis,’ and it should be
noted that the null hypothesis is never proved or
established, but is possibly disproved, in the course
of experimentation. Every experiment may be said
to exist only in order to give the facts a chance of
disproving the null hypothesis.”
    H1 is important hypothesis: the research
   hypothesis is what we hope to show is true
      Asymmetry: cannot infer H0 is true but
        can reject H0 and infer H1 is true         9
      Type I and Type II Errors
• Type I Error: Reject a
  true null hypothesis                 Guilty    Innocent
   – Innocent defendant
     wrongly convicted
   – DNA tests exonerate                No        Type I
                             Convict
• Type II Error: Fail to               Error      Error
  reject a false null hyp.
   – Guilty defendant
     wrongly set free                  Type II     No
                             Acquit
                                        Error     Error
   – OJ Simpson?
                                                           10
      Type I and Type II Errors
• P(Type I Error) = α
• α is the significance                Guilty    Innocent
  level of the statistical
  test
• P(Type II Error) = β       Convict
                                        No        Type I
                                       Error      Error
• What is probability
  jury makes no errors
  (reaches the right                   Type II     No
                             Acquit
  conclusion)?                          Error     Error

                                                           11
       Statistician Chooses α
• Researcher chooses significance level (α)
  – Researcher selects the probability of Type I
    error that is tolerable for the question at hand
• For example, if a researcher chooses a
  conventional significance level α = 0.05
  – 5% of the time the researcher will reject a true
    null hypothesis
     • What kind of error is this: Type I or Type II?
     • What can be done to reduce this error?
                                                        12
                  α and β
• Decreasing α (probability of Type I error)
  increases β (probability of Type II error)
  – If raise burden of proof (decrease α) so as not
    to convict an innocent person we increase the
    chance we let a guilty one go free (increase β)
  – If we lower the burden of proof (increase α) so
    that we can “put criminals behind bars”
    (decrease β) we increase the chance we send
    innocent people to jail

                                                 13
      Setting Up Hypotheses
• Inference about unknown parameter θ :
  – H0: θ   = some number (Ex. H0: θ   = 25)
  – H1: θ   ≠ some number (Ex. H1: θ   ≠ 25)
  – H1: θ   > some number (Ex. H1: θ   > 25)
  – H1: θ   < some number (Ex. H1: θ   < 25)
• Show that the demand for apples is elastic
  – H0:
  – H1:

                                               14
         Ex: Hotel Profit Margins
• Economist wants to                          Histogram of Hotel Data
  show hotel industry                                 n = 25




                                 0 5 101520
  profit margins are on




                                   Density
  average negative in
  April 2005
    – H0: μ = 0                               -.12       -.06         0
    – H1: μ < 0                               April 2005 Profit Margin

• Random sample: 25
                                      What should we infer?
  hotels and ask each
  April profit margins             In this example do we need
                                         formal inference?
This is a hypothetical example                                       15

								
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