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174 CHAPTER 6 ESTIMATION
Normal distribution
t distribution
FIGURE 6.3.2
Comparison of normal distribution and t distribution.
x
6. Compared to the normal distribution, the t distribution is less peaked in the center
and has thicker tails. Figure 6.3.2 compares the t distribution with the normal.
7. The t distribution approaches the normal distribution as n - 1 approaches infinity.
The t distribution, like the standard normal, has been extensively tabulated. One
such table is given as Table E in the Appendix. As we will see, we must take both the
confidence coefficient and degrees of freedom into account when using the table of the
t distribution.
You may use MINITAB to graph the t distribution (for specified degrees-of-freedom
values) and other distributions. After designating the horizontal axis by following directions
in the Set Patterned Data box, choose menu path Calc and then Probability Distributions.
Finally, click on the distribution desired and follow instructions. Use the Plot
dialog box to plot the graph.
Confidence Intervals Using t The general procedure for constructing confidence
intervals is not affected by our having to use the t distribution rather than the
standard normal distribution. We still make use of the relationship expressed by
estimator ; 1reliability coefficient2 * 1standard error of the estimator2
What is different is the source of the reliability coefficient. It is now obtained from the
table of the t distribution rather than from the table of the standard normal distribution.
To be more specific, when sampling is from a normal distribution whose standard deviation,
s, is unknown, the 10011 - a2 percent confidence interval for the population
mean, m, is given by
s
x ; t 11-a>22
(6.3.2)
1n
We emphasize that a requirement for the strictly valid use of the t distribution is that the
sample must be drawn from a normal distribution. Experience has shown, however, that
moderate departures from this requirement can be tolerated. As a consequence, the t distribution
is used even when it is known that the parent population deviates somewhat
from normality. Most researchers require that an assumption of, at least, a mound-shaped
population distribution be tenable.
EXAMPLE 6.3.1
Maffulli et al. (A-1) studied the effectiveness of early weightbearing and ankle mobilization
therapies following acute repair of a ruptured Achilles tendon. One of the variables