Hypothesis_Testing by fanzhongqing

VIEWS: 3 PAGES: 6

									                     A Paradigm for Hypothesis Testing

Statistical hypothesis testing always follows the same series of steps:

A    Formulate the null hypothesis (the statement "on trial"). As the following
     discussion will show, if your ultimate goal is to conclude that you have
     evidence supporting a claim, you must take the opposite of that claim as
     your null hypothesis.

B    Look at your data, and find the version of the null hypothesis which comes
     closest to fitting your data. (Some null hypotheses have only one version, but
     others may be "true" in many different specific ways. This step is analogous
     to "giving the accused the benefit of the doubt" in a criminal trial.)

C    Determine what you would have "expected" your study to yield, had it been
     performed in a world where this "fitted" version of the null hypothesis was true.

D    Measure how "different" your actual study result is from this expectation.

E    Compute the probability that a study such as yours, conducted in a world
     where the fitted version of the null hypothesis really is true, would - just by
     chance (i.e., due to sampling error) - yield a difference this large or larger,
     in a direction that contradicts the null hypothesis. This probability is the
     significance level of your data, with respect to the null hypothesis.

F1   If the significance level is a "large" percentage, then you conclude that the
     data does not provide meaningful evidence against the null hypothesis. You've
     found that, in a world where the statement is true, you'd frequently see
     evidence "like" what you're seeing. Note that you don't conclude that the data
     supports the null hypothesis - only that it doesn't provide much of a
     contradiction.

F2   If, on the other hand, the significance level is near 0, then you know that
     either (1) you were very unlucky when you collected your data, and obtained
     a very misrepresentative sample, or (2) the null hypothesis is false. Since
     you don't expect to be very unlucky on a regular basis, your data, all by
     itself, makes you very suspicious. You are entitled to say that your data
     strongly contradicts the null hypothesis, and strongly supports the alternative
     (the opposite of the null hypothesis).

     A personal interpretation of the numerical significance level into words
     expressing the "strength" of the evidence can be generated by playing
     through our "coin-flipping" exercise.
              A Paradigm for Hypothesis Testing, Applied

Consider our example involving the commercial loan officer of a bank. The credit
manager claimed that the mean balance due on customer accounts was at least
$300. We took a sample of 64 customers, and found a sample mean of $280,
with a sample standard deviation of $120.

A   Formulate the null hypothesis (the statement "on trial"). As the following
    discussion will show, if your ultimate goal is to conclude that you have
    evidence supporting a claim, you must take the opposite of that claim as
    your null hypothesis.

    We're not out to "prove" a statement of our own here. The credit manager of
    the firm seeking a loan has made a claim: "The mean balance due (and
    soon to be paid) on customer credit accounts is at least $300." This is our
    null hypothesis.

B   Look at your data, and find the version of the null hypothesis which comes
    closest to fitting your data. (Some null hypotheses have only one version, but
    others may be "true" in many different specific ways. This step is analogous
    to "giving the accused the benefit of the doubt" in a criminal trial.)

    Our study yielded a sample mean of $280. The precise version of the null
    hypothesis that comes closest to matching this is that the mean balance
    due is exactly $300.

C   Determine what you would have "expected" your study to yield, had it been
    performed in a world where this "fitted" version of the null hypothesis was true.

    We would have expected a sample mean of $300.

D   Measure how "different" your actual study result is from this expectation.

    Our actual result is below this expectation by $20.

E   Compute the probability that a study such as yours, conducted in a world
    where the fitted version of the null hypothesis really is true, would - just by
    chance (i.e., due to sampling error) - yield a difference this large or larger,
    in a direction that contradicts the null hypothesis. This probability is the
    significance level of your data, with respect to the null hypothesis.

    To contradict the null hypothesis by this much or more, we would need to have
    obtained a sample mean of $280 or less.

    One standard-deviation's-worth of "fuzz" (more precisely, "exposure to
    sampling error") in our estimation procedure is

        $15      #NAME?

    If the true population mean is $300, and a standard-deviation's-worth of
    uncertainty in our estimate is $15, then the chance that - just due to bad
    luck - we'd get an estimate of $280 or less is
       9.12%      #NAME?
       9.36%      #NAME?

     This is the significance level of our data, with respect to the null hypothesis.

F1   If the significance level is a "large" percentage, then you conclude that the
     data does not provide meaningful evidence against the null hypothesis. You've
     found that, in a world where the statement is true, you'd frequently see
     evidence "like" what you're seeing. Note that you don't conclude that the data
     supports the null hypothesis - only that it doesn't provide much of a
     contradiction.

F2   If, on the other hand, the significance level is near 0, then you know that
     either (1) you were very unlucky when you collected your data, and obtained
     a very misrepresentative sample, or (2) the null hypothesis is false. Since
     you don't expect to be very unlucky on a regular basis, your data, all by
     itself, makes you very suspicious. You are entitled to say that your data
     strongly contradicts the null hypothesis, and strongly supports the alternative
     (the opposite of the null hypothesis).

     A personal interpretation of the numerical significance level into words
     expressing the "strength" of the evidence can be generated by playing
     through our "coin-flipping" exercise.

     Personally, I'd interpret this as "a bit of evidence against the null hypothesis."
        A Paradigm for Hypothesis Testing (Yttrium.xls, 13 and 14)

A      Formulate the null hypothesis (the statement "on trial"). As the following
       discussion will show, if your ultimate goal is to conclude that you have
       evidence supporting a claim, you must take the opposite of that claim as
       your null hypothesis.

    13 average time spent online by subscribers is ≤ 800 minutes/month
       m time  800

    14 average increase in time online associated with an additional $1000 in monthly salary is ≤ 70 minutes
       1000*coef income  70 (in the most complete model, since we're dealing with an "effect" here)

       In both cases, we wish to make an affirmative assertion, so we need to take the
       opposite as our null hypothesis.

B      Look at your data, and find the version of the null hypothesis which comes
       closest to fitting your data. (Some null hypotheses have only one version, but
       others may be "true" in many different specific ways. This step is analogous
       to "giving the accused the benefit of the doubt" in a criminal trial.)              Univariate statistics

    13 m time = 800                                                                        mean
                                                                                           standard deviation
    14 1000*coef income = 70                                                               standard error of the mean

                                                                                           Regression: time

                                                                                           coefficient
C      Determine what you would have "expected" your study to yield, had it been           std error of coef
       performed in a world where this "fitted" version of the null hypothesis was true.

    13 a sample mean of 800

    14 an estimated coefficient of 0.07

D      Measure how "different" your actual study result is from this expectation.

    13 22.67 above

    14 0.007889 above

E      Compute the probability that a study such as yours, conducted in a world
       where the fitted version of the null hypothesis really is true, would - just by
       chance (i.e., due to sampling error) - yield a difference this large or larger,
       in a direction that contradicts the null hypothesis. This probability is the
       significance level of your data, with respect to the null hypothesis.

    13 11.195%        #NAME?
       11.339%        #NAME?

       Only sample means above 822.57 would be at least this contradictory, so we
      use the upper tail of the normal distribution.

 14    3.268%      #NAME?
       3.268%      #NAME?
       3.422%      #NAME?

F1    If the significance level is a "large" percentage, then you conclude that the
      data does not provide meaningful evidence against the null hypothesis. You've
      found that, even in a world where the statement is true, you'd frequently see
      evidence "like" what you're seeing. Note that you don't conclude that the data
      supports the null hypothesis - only that it doesn't provide much of a
      contradiction.

F2    If, on the other hand, the significance level is near 0, then you know that
      either (1) you were very unlucky when you collected your data, and obtained
      a very misrepresentative sample, or (2) the null hypothesis is false. Since
      you don't expect to be very unlucky on a regular basis, your data, all by
      itself, makes you very suspicious. You are entitled to say that your data
      strongly contradicts the null hypothesis, and strongly supports the alternative
      (the opposite of the null hypothesis).

      A personal interpretation of the numerical significance level into words
      expressing the "strength" of the evidence can be generated by playing
      through our "coin-flipping" exercise.

 13 only a little bit of evidence supporting the desired claim

 14 strong (but not extremely strong) evidence supporting the desired claim
                       time
                        822.67
                     186.3936
error of the mean    18.63936


            constant   sex      age     income
              65.259 32.4438     6.8591 0.077889
            25.01844 11.23912 0.699278 0.004281

								
To top