84 ESTIMATION OF POPULATION MEANS: CONFIDENCE INTERVALSFigure 7.2 Normal distribution compared with t distributions.be estimated by computing the sample standard deviation s. This requires use of the
quantity t = (7 - p)/(s/&), which will be used instead of z = (7 - p)/(o/&).
The distribution of t (often called Student’s t distribution) does not quite have
a normal distribution. The t distribution is bell-shaped and symmetric, as is the
normal distribution, but is somewhat more widely spread than the standard normal
distribution. The distribution oft differs for different values of n, the sample size;
if the sample size is small, the curve has more area in the “tails”; if the sample size
is large, the curve is less spread out and is very close to the standard normal curve.
Figure 7.2 shows the normal distribution and the t distributions for sample sizes n = 3
and n = 5.
The areas under the distribution curves to the left oft have been computed and
put in table form; Table A.3 gives several areas from --co to t[X] for some of these
distributions. The first column of Table A.3 lists a number called the degrees of
freedom (d.f.’s); this is the number that was used in the denominator in the calculation
of s2, or n - 1.
We need to know the degrees of freedom whenever we do not know u2. If we knew
the population mean p, the estimate of u would be c(X - ~)~/n and the d.f.’s would
be n. When the population mean is unknown, we estimate it by the sample mean
X, thus limiting ourselves to samples that have 5? as their sample mean. With such
samples, if the first n - 1 observations are chosen, the last observation is determined
in such a way that the mean of the n observations is x. Thus we say that there are
only n - 1 d.f. in estimating s2.
In the column headings of Table A.3 are the areas from -m to t[X], the numbers
below which A% of the area lies. The row headings are d.f., and in the body of Table
A.3, t[X] is given. For example, with 6 d.f. under X = .90, we read t[.90] = 1.440;
90% of the area under the t distribution lies to the left of 1.440. For each row of the
table, the values of t[X] are different depending on the d.f.’s. Note that for d.f. = x
(infinity), the t distribution is the same as the normal distribution. For example, for
X = .975 andd.f. = cc,t[.975] = 1.96. Also z[.975] = 1.96 fromTableA.2.