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  of 30.77 and an  of 4.43 (  = 19.58)
If we repeatedly sampled groups of 6 people, found the  of their estimates, and then plotted the  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 H 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 H_A 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

 = 30.77     Critical = 55 

 = 4.43     ( = 19.58) 

  

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 
Before setting some level of  it is important to realize that levels of  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 
Unfortunately, it is impossible to precisely calculate  because we do not know the shape of the sampling distribution under H₁
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

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)

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  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