## Degrees of freedom and t tables

Table 2-1 is an abbreviated t score table that shows the values of t corresponding to different areas under the normal distribution for various sample sizes. Tables of t values do not show sample size (n) directly; instead, they express sample size in terms of degrees of freedom (df). For the purposes of USMLE, degrees of freedom (df) can be defined as simply equal to n — 1. Therefore, ro determine the value of t (such that 95% of the population of t-statistics based on a sample size of 15 lies between — t and +t), one would look in the table for the appropriate value of t for df = 14 (14 being equal to n — 1); this is sometimes written as t,4. Table 2-1 shows that this value is 2.145.

As n becomes larger (100 or more), the values of t are very close to the corresponding values of z- As the middle column shows, for a df of 100, 95% of the distribution falls within t = ± 1.984; while for a df of this figure is 1.96, which is the same figure for z (see Table 1-3). In general, the value of t that divides the central 95% of the distribution from the remaining 5% is in the region of 2, just as it is for (One- and two-tailed tests are discussed in Chapter 3 in the section on Directional Hypotheses).

As an example of the use of t scores, we can repear the earlier task of estimating (with 95% confidence) the true mean resting heart rate of a large population, basing the estimate on a random sample of people drawn from this population. This time we will not make the unrealistic assumption that the standard error is known.

As before, a random sample of 15 people is drawn, and it is found that their mean heart rate (X) is 74 beats/min. Assuming that the standard deviation of this sample is 8.2, the estimated standard error, can be calculated as follows: 