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Saturday, 4 May 2013

Elementary Statistics chapter 5


Sampling Distributions and Hypothesis Testing



Statistics is arguing

    Typically, we are arguing either 1) that some value (or mean) is different from some other mean, or 2) that there is a relation between the values of one variable, and the values of another.
    Thus, following Steve's in-class example, we typically first produce some null hypothesis (i.e., no difference or relation) and then attempt to show how improbably something is given the null hypothesis.

Sampling Distributions

    Just as we can plot distributions of observations, we can also plot distributions of statistics (e.g., means)
    These distributions of sample statistics are called sampling distributions
    For example, if we consider the 48 students in my class who estimated my age as a population, their guesses have a ch4-1 of 30.77 and an ch4-2 of 4.43 ( ch4-3 = 19.58)
    If we repeatedly sampled groups of 6 people, found the ch4-4 of their estimates, and then plotted the ch4-5 s, the distribution might look like

Hypothesis Testing

    What I have previously called "arguing" is more appropriately called hypothesis testing
    Hypothesis testing normally consists of the following steps:
    • some research hypothesis is proposed (or alternate hypothesis) - H1
    • the null hypothesis is also proposed - H0
    • the relevant sampling distribution is obtained under the assumption that H0 is correct
    • I obtain a sample representative of H1 and calculate the relevant statistic (or observation)
    • Given the sampling distribution, I calculate the probability of observing the statistic (or observation) noted in step 4, by chance
    • On the basis of this probability, I make a decision

The beginnings of an example

    One of the students in our class guessed my age to be 55. I think that said student was fooling around. That is, I think that guess represents something different that do the rest of the guesses
    H0 - the guess is not really different
    H1 - the guess is different
    • obtain a sampling distribution of 0
    • calculate the probability of guessing 55, given this distribution
    • Use that probability to decide whether this difference is just chance, or something more

A Touch of Philosophy

    Some students new to this idea of hypothesis testing find this whole business of creating a null hypothesis and then shooting it down as a tad on the weird side, why do it that way?
    This dates back to a philosopher guy named Karl Popper who claimed that it is very difficult to prove something to be true, but no so difficult to prove it to be untrue
    So, it is easier to prove H0 to be wrong, than to prove HA to be right
    In fact, we never really prove H1 to be right. That is just something we imply (similarly H0)

Using the Normal Distribution to Test Hypotheses

    The "Steve's Age" example begun earlier is an example of a situation where we want to compare one observation to a distribution of observations
    This represents the simplest hypothesis-testing situation because the sampling distribution is simply the distribution of the individual observations
    Thus, in this case we can use the stuff we learned about z-scores to test hypotheses that some individual observation is either abnormally high (or abnormally low)
    That is, we use our mean and standard deviation to calculate the a z-score for the critical value, then go to the tables to find the probability of observing a value as high or higher than (or as low or lower than) the one we wish to test
Finishing the example
ch4-6 = 30.77     Critical = 55 
ch4-7 = 4.43     (ch4-8 = 19.58) 
ch4-9 ch4-10 ch4-11
    From the z-table, the area of the portion of the curve above a z of 3.21 (i.e., the smaller portion) is approximately .0006
    Thus, the probability of observing a score as high or higher than 55 is .0006.

Making decisions given probabilities

    It is important to realize that all our test really tells us is the probability of some event given some null hypothesis
    It does not tell us whether that probability is sufficiently small to reject H0, that decision is left to the experimenter
    In our example, the probability is so low, that the decision is relatively easy. There is only a .06% chance that the observation of 55 fits with the other observations in the sample. Thus, we can reject H0 without much worry
    But what if the probability was 10% or 5%? What probability is small enough to reject H0?
    It turns out there are two answers to that:
    • the real answer
    • the "conventional" answer

The Real Answer -

or Type I and Type II errors

    First some terminology...
    The probability level we pick as our cut-off for rejecting H0 is referred to as our rejection level or our significance level
    Any level below our rejection or significance level is called our rejection region
    OK, so the problem is choosing an appropriate rejection level
    In doing so, we should consider the four possible situations that could occur when we're hypothesis testing
      H0 true H0 false
      Reject H0 Type I error Correct
      Fail to Correct Type II error
      Reject H0Type I Error
Type I Error
    Type I error is the probability of rejecting the null hypothesis when it is really true
    e.g., saying that the person who guessed I was 55 was just screwing around when, in fact, it was an honest guess just like the others
    We can specify exactly what the probability of making that error was, in our example it was .06%
    Usually we specify some "acceptable" level of error before running the study
    • then call something significant if it is below this level
    This acceptable level of error is typically denoted as ch4-12
    Before setting some level of ch4-13 it is important to realize that levels of ch4-14 are also linked to type II errors
Type II Error
    Type II error is the probability of failing to reject a null hypothesis that is really false
    e.g., judging OJ as not guilty when he is actually guilty
    The probability of making a type II error is denoted as ch4-15
    Unfortunately, it is impossible to precisely calculate ch4-16 because we do not know the shape of the sampling distribution under H1
    It is possible to "approximately" measure ch4-17 , and we will talk a bit about that in Chapter 8
    For now, it is critical to know that there is a trade-off between ch4-18 and ch4-19 , as one goes down, the other goes up
    Thus, it is important to consider the situation prior to setting a significance level
The "Conventional" Answer
    While issues of type I versus type II error are critical in certain situations, psychology experiments are not typically among them (although they sometimes are)
    As a result, psychology has adopted the standard of accepting ch4-20 =.05 as a conventional level of significance
    It is important to note, however, that there is nothing magical about this value (although you wouldn't know it by looking at published articles)

One versus Two Tailed Tests

    Often, we are interested in determining if some critical difference (or relation) exists and we are not so concerned about the direction of the effect
    That situation is termed two-tailed, meaning we are interested in extreme scores at either tail of the distribution
    Note, that when performing a two-tailed test we must only consider something significant if it falls in the bottom 2.5% or the top 2.5% of the distribution (to keep ch4-21 at 5%)
    If we were interested in only a high or low extreme, then we are doing a one-tailed or directional test and look only to see if the difference is in the specific critical region encompassing all 5% in the appropriate tail
    Two-tailed tests are more common usually because either outcome would be interesting, even if only one was expected

Other Sampling Distributions

    The basics of hypothesis testing described in this chapter do not change
    All that changes across chapters is the specific sampling distribution (and its associated table of values)
    The critical issue will be to realize which sampling distribution is the one to use in which situation
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