Using the Student's t-distribution

Since you are proficient at finding confidence intervals for normal distributions using the z* table or calculator commands it will be a relatively smooth transition to the t-distribution which looks "almost" normal. The distribution is symmetric but with slightly different areas under the curve that appear as a thickening of the tails.

t model

There are many t-distributions which vary according to sample size. Of course, if the sample were really large, like infinite, the t-distribution and the Normal distribution would merge. As the sample size and associated degrees of freedom values INCREASE, the t-distribution begins to approach Normal.

There is a new descriptor for this distribution called "degrees of freedom." The best analogy for degrees of freedom is to THINK about solving a multi-variable formula, something like area of a rectangle = Length * Width or A = LW. Having THREE variables REQUIRES us to KNOW TWO of them, or n - 1, in order to solve for the remaining variable. Two of them are FREE to take on various values but the remaining variable MUST make the statement true. The last variable has NO FREEDOM. Its value is determined based on the other two. Similarly, if we had a complex 9-variable formula, 8 of the variables would be FREE to vary while the remaining variable would NOT. IF we could theorize INFINITE degrees of freedom...the t distribution would be EXACTLY Normal.

Another characteristic of the t-distribution is that it can withstand deviations from the conditions necessary for inference without much loss of accuracy in the answer. The official term to describe this is "robust." When small samples are involved it is always wise to use the t-distribution and the t* table of areas. Always consult the legend when using a new table of values.

Be aware that many verbs can be used to ask the same basic question.

CONSTRUCT a confidence interval
CREATE a confidence interval
DETERMINE a confidence interval
FIND a confidence interval

Once the interval of values has been determined the result must be analyzed and interpreted for meaning in the context of the problem. Please see the Body Temperatures Example in the sidebar under HAND OUTS to see how the calculator can be used to answer confidence interval questions.

More detail and examples are contained in the lesson below. This lesson explains the basics of using the t-distribution. Click on the lesson to download.

T-Distributions

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