Part III: One & Two Sample Parametric Tests
One Sample t and z Tests
| One Sample z Test | One Sample t Test | But What are Degrees of Freedom |
 
 
One sample tests, in general, examples of "goodness of fit" tests where we are testing whether our data supports predictions regarding the value of the population mean. This is most commonly represented as follows:
Ho: m = mo where m = actual true population mean
mo = hypothesized population mean (under Ho)

In order to calculate a z-score, you would need to use the following formula:

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Why do we use the standard error? Z is a test for the sample mean. To test Ho, we want to evaluate the probability or likelihood of getting our result, x, if Ho is true. To do this, we need to see where x falls on its sampling distribution. The z statistic is the provides us with this.

Now if the population standard deviation is unknown, but N is large (N >= 30), you can still use z, but you must substitute the sample standard devation for the population standard deviation when calculating the standard error of the mean.
 
 
The t-test is a test of the same null hypothesis as above. However, this time we have a new index of departure and, hence, a new formula to learn. In small samples (N < 30), sample standard deviations are biased estimates of their corresponding population standard deviations. In other words, s does not estimate s perfectly -- it is usually to small. Thus, we simply adjust the standard error using something called degrees of freedom (in this case, df = N - 1). Substituting these variations into the traditional z formula, we obtain the t test formula:

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The concept is most easily explained in the context of one-sample tests; that is, in cases were we are comparing some group to a known population mean. In the past, we've dealt with z-scores. Recall that z-scores only had one distribution, and we used it to get the proportions/probabilities of various things. The really nice thing about the z-distribution is that it is normally distributed. This means that whenever we do some significance test--and we know the population standard deviation--we can calculate z-scores and easily find the probabilities associated with them. Once we've done this, we're one simple logical step away from rejecting or failing to reject our null hypothesis.

But what happens when we don't know the population standard deviation? We can only do our significance testing if we estimate our population standard deviation from whatever we find in our sample. And when we do this, we need to worry about a new sampling distribution.

The sampling distribution of t-scores (the t-distribution) is really a family of distributions. This means that there are many sampling distributions associated with t-scores. These distributions, in general, are unimodal (one mode), symmetric, and bell-shaped. For the most part, they look kind of like the traditional z-distribution; however, they tend to be, according to Hays, "a little 'plumper' in the extreme regions and slightly 'flatter' in the central region." However, when the sample size gets really big, the curves gradually start to look more and more "normal."

The thing that determines the shape of the distribution is something called the degrees of freedom. By definition, degrees of freedom are the number of scores that are free to vary. Yeah, but how does that explain anything? How are these degrees of freedom calculated?

Remember that standard deviation is derived from the sample variance, and that the variance is calculated using the sum of squared deviations from the mean of your sample. This meant that we subtracted the mean from each score, squared them, and then added them all up. Anyone remember why we squared the deviation scores? Anyone? If we didn't square them before we added them up, the total sum would have equaled zero (double check all your old notes!). Okay, so keep this in mind for a few minutes.

Suppose I tell you that some sample has four scores (N=4) and that I ask you to guess how the four students did on some test that I gave them. So you start a guessin' their deviation scores . . . student number one gets a 6, student number two gets a -9, student number three gets a -7. But what happens when you have to guess the fourth score? Remember, all deviation scores have to add up to zero. Therefore, based on your first three guesses, your fourth guess has to be 10 (6 + -9 + -7 + 10 = 0). Thus, you were free to guess any number you wanted to for the first three deviation scores but the fourth score was determined by the first three scores--the fourth score was not free to vary!

What this means is that deviation scores are not entirely independent of each other. This wasn't a problem when we knew the population parameters. But now we have to estimate them. So, to make our estimates "unbiased" (i.e. more accurate), we divide the sum of squares by the degrees of freedom instead of the sample size to get a number for the variance. This tends to increase the variance, thus accounting for the fact that a "biased" estimate underestimates the variance.

This concept gets a little uglier as we more through increasingly more complicated statistical tests. We will have different formulae for different tests, which will have different sampling distributions, and will in turn have their own formulae for degrees of freedom. But don't get too frazzled, the basic concepts stay the same: 1) the shape of a non-normal sampling distribution is determined by the number of degrees of freedom, and 2) the degrees of freedom are used to more accurately estimate a population parameter in those cases when it is not known.