t-curves
1) When we work with z-scores, we are working with one curve, namely, The standard normal curve.
2) Using this curve (µ=0, s=1), we can find critical z-values that are at the boundaries of the curve at both tails when a given confidence level is specified.
3) For example, the popular 95% C-Level has critical z-values of 1.96 at the right tail and -1.96 at the left tail. The 95% indicates the amount of area covered under the curve and over the confidence
interval. The 5% uncovered area is split over both tails and is known as the significance level a.
4) To find these critical values by our calculator (TI-83), we go to 2nd, VARS, menu 3 and enter the total area to the left of the critical value we are seeking. In the 95% C-Level, we would enter 1-.05/2. Generally, we would enter (1-a/2), where a is specified by the C-Level.
5) For finding the probabilities (Areas) between to z-scores, we would go to 2nd, VARS, menu 2 and enter the left z-score, then the right z-score.
6) Think of a t-curve as one of many standard normal curves (each one varying, depending on sample size n) The phrase, “degrees of freedom” is defined as n-1. (sample size minus one). (the number of quantities that may be assigned any value to get a desired result. The nth quantity would be determined).
For example: Take any 4 numbers and want their mean to equal 20. You may assign values to the first 3, say, 2, 5, 7, but not the 4th for an average of 20. The 4th number must be 6 (after assigning the first three) for the desired mean of 20. So, in this case, there are n in our case, but n-1 are free to take on any value (degrees of freedom), but not all 4.
7) When we work with t-curves, we are working with an infinite number of normal curves, not just one.
The reason for this is the fact that the population standard deviation is unknown and we are using the sample standard deviation as an estimate of the population standard deviation. But, we know from a previous lesson, that for means, the relationship between s (population standard deviation) and the sample standard deviation is given by: Sx=s/√n, n is the size of the sample. (Assuming n<.05N, where N is the size of the population). If not, then the finite population correction factor must be used as a factor on the right side. (see link on my web site). So, the larger the n, the smaller Sx becomes.
8) So, each t-curve has a different standard deviation depending upon the size of the sample n.
For small samples (smaller n), they are wider, then become narrower as n increases.
9) So, calculating critical values for t is not as easy as it was for z-values (since we have many to consider).
10) A table has been formulated that has the popular values present corresponding to select sample sizes.
The left column indicates the degrees of freedom. (i.e, n-1). The headings are various values of a.
11) The TI-83 does not have a similar menu for finding these like in step 4 above, however, there is an indirect way of getting them. Here’s how: Go to STAT, Tests, menu 8 , insert a mean =0, Sx=√n , n, C-Level. The t interval will be displayed with the critical values at the end. These entries are based on the formula for a t value with a given C-level and df(degrees of freedom).
12) How about the reverse problem of finding areas (probabilities) under a t-curve. This can be done by the TI-83 calculator, as well. Go to 2nd, VARS, shade in DISTR, go down to tcdf, enter (t value at the left boundary, following by the one at the right boundary, followed by the degrees of freedom). The result will be the area under that t-curve between those two values.
Note: Many statisticians do not use t-curves, just the standard normal curve (z-scores) since their results do not differ significantly with larger sample sizes. However, with smaller sample sizes, most studies give significant differences and opt to use t-curves for more accuracy.