Psy B07 Chapter 4Slide 1 SAMPLING DISTRIBUTIONS AND HYPOTHESIS TESTING

Psy B07 Chapter 4Slide 2  Sampling Distributions revisited  Hypothesis Testing  Using the Normal Distribution to test Hypotheses  Type I and Type II Errors  One vs. Two Tailed Tests Outline

Psy B07 Chapter 4Slide 3 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, 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.

Psy B07 Chapter 4Slide 4 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 24 students in a tutorial who estimated my weight as a population, their guesses have an x of and an  of (  2 = )

Psy B07 Chapter 4Slide 5 Sampling Distributions  If we repeatedly sampled groups of 6 people, found the x of their estimates, and then plotted the x’s, the distribution might look like:

Psy B07 Chapter 4Slide 6 Hypothesis Testing  What I have previously called “arguing” is more appropriately called hypothesis testing.  Hypothesis testing normally consists of the following steps: 1) some research hypothesis is proposed (or alternate hypothesis) - H 1. 2) the null hypothesis is also proposed - H 0.

Psy B07 Chapter 4Slide 7 Hypothesis Testing 3) the relevant sampling distribution is obtained under the assumption that H 0 is correct. 4) I obtain a sample representative of H 1 and calculate the relevant statistic (or observation). 5) Given the sampling distribution, I calculate the probability of observing the statistic (or observation) noted in step 4, by chance. 6) On the basis of this probability, I make a decision

Psy B07 Chapter 4Slide 8 The Beginnings of an Example  One of the students in the tutorial guessed my weight to be 200 lbs. I think that said student was fooling around. That is, I think that guess represents something different that do the rest of the guesses.  H 0 - the guess is not really different.  H 1 - the guess is different.

Psy B07 Chapter 4Slide 9 The Beginnings of an Example 1) obtain a sampling distribution of H 0. 2) calculate the probability of guessing 200, given this distribution 3) Use that probability to decide whether this difference is just chance, or something more.

Psy B07 Chapter 4Slide 10 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 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.

Psy B07 Chapter 4Slide 11 A Touch of Philosophy  So, it is easier to prove H 0 to be wrong, than to prove H A to be right.  In fact, we never really prove H 1 to be right. That is just something we imply (similarly H 0 ).

Psy B07 Chapter 4Slide 12 Using the Normal Distribution to test Hypotheses  The “Marty’s Weight” 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.

Psy B07 Chapter 4Slide 13 Using the Normal Distribution to test Hypotheses  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.

Psy B07 Chapter 4Slide 14 Finishing the Example  = Critical = 200  = 12.43

Psy B07 Chapter 4Slide 15 Finishing the Example  From the z-table, the area of the portion of the curve above a z of 2.51 (i.e., the smaller portion) is approximately  Thus, the probability of observing a score as high or higher than 200 is.0060

Psy B07 Chapter 4Slide 16 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 H 0, 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.60% chance that the observation of 200 fits with the other observations in the sample. Thus, we can reject H 0 without much worry.

Psy B07 Chapter 4Slide 17 Making Decisions given Probabilities  But what if the probability was 10% or 5%? What probability is small enough to reject H 0 ?  It turns out there are two answers to that:  the real answer.  the “conventional” answer.

Psy B07 Chapter 4Slide 18 The “Real” Answer  First some terminology....  The probability level we pick as our cut-off for rejecting H 0 is referred to as our rejection level or our significance level.  Any level below our rejection or significance level is called our rejection region

Psy B07 Chapter 4Slide 19 The “Real” Answer  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.

Psy B07 Chapter 4Slide 20 Type I and Type II Errors  Type I error is the probability of rejecting the null hypothesis when it is really true.  Example: saying that the person who guessed I weigh 200 lbs 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.60%.

Psy B07 Chapter 4Slide 21 Type I and Type II Errors  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   Before setting some level of it is important to realize that levels of  are also linked to Type II errors

Psy B07 Chapter 4Slide 22 Type I and Type II Errors  Type II error is the probability of failing to reject a null hypothesis that is really false.  Example: judging OJ as not guilty when he is actually guilty.  The probability of making a Type II error is denoted as 

Psy B07 Chapter 4Slide 23 Type I and Type II Errors  Unfortunately, it is impossible to precisely calculate  because we do not know the shape of the sampling distribution under H 1.  It is possible to “approximately” measure , 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  and , as one goes down, the other goes up.  Thus, it is important to consider the situation prior to setting a significance level.

Psy B07 Chapter 4Slide 24 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 =.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).

Psy B07 Chapter 4Slide 25 One vs. Two Tailed Tests  Often, we want to determine 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  at 5%).

Psy B07 Chapter 4Slide 26 One vs. Two Tailed Tests  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.

Psy B07 Chapter 4Slide 27 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.