Statistical Methods

The problem with z-scores as an inferential test is that the formula requires more information that is usually available—usually, we will not know the value of the population standard deviation, which is needed to compute the standard error.

For situations where the population standard deviation is not known, we use the t statistic rather than a z-score.

When the variability for the population is not known, we use the sample variability in its place.

Now that we have computed an estimate of the standard error, we can replace the population parameter in the z-score formula to get the t statistic formula:

The exact shape of the t distribution changes with degrees of freedom.

In fact, statisticians speak of a "family" of distributions.

As the df gets very large, the shape of the t distribution gets closer in shape to the normal z distribution.

Just as we used the unit normal table to locate proportions associated with z-scores, we will use a t-distribution table to find proportions for t statistics.

See page 205 in the text: The two rows at the top of the table show proportions of the t distribution contained in either one or two tails, depending on which row is used.

The first column of the table lists degrees of freedom for the t statistic.

The numbers in the body of the table are the t values that mark the boundary between the tails and the rest of the t-distribution.

Example: with df = 3, exactly 5% of the t-distribution is located in the tail beyond t = 2.353

Example: the table lists values for df = 40 and df = 60, but does ont list any entries for df values between 40 and 60.

In this situation, you should look up the critical values of t for both of the surrounding df values and then use the larger value for t.

If your sample t statistic is greater than the larger value listed, you can be certain that the data are in the critical region and you can confidently reject the null.

The big difference is in STEP 2: Locating the Critical Region. For t tests, you are using the t statistic rather than z, and therefore must use the t-distribution table when determining the critical region.

These are expressed in terms of population parameters

H₀:μ = 30

H₁: μ ≠ 30

Alpha = .05

First, we must obtain the degrees of freedom:

df = n – 1 = 16 – 1 = 15

If we examine the table, when alpha = .05 and df = 15, then t_critical = 2.131

A. Calculate the sample variance (SS = 1215, df = 15)

This works like z did: if your computed value is more than your critical value, you can reject the null hypothesis.

The values in the sample must consist of independent observations—there is no systematic relationship between the sample elements (this means it is inappropriate if the same subjects are measured twice)

The population sampled should be normal—this is not very important for relatively large sample sizes—the test is robust against this violation.