Basic Statistics

CONFIDENCE INTERVAL FORTHE MEAN USING THE t DISTRIBUTION 85

7.4 CONFIDENCE INTERVAL FOR THE MEAN USING THE

t DISTRIBUTION

When the standard deviation is estimated from the sample, a confidence interval for
p, the population mean, is formed just as when o is known, except that s now replaces
o and the t tables replace the normal tables. With s calculated from the sample, a
95% confidence interval is

• _. t1.9751~
  xk-

The t[.975] denotes the value below which 97.5% of the t’s lie in the t distribution
with n - 1 d.f.
The quantity slfi is usually called the standard error of the mean in computer
program output. Minitab, SAS, and Stata compute confidence limits, and SPSS
provides plots of the mean and the confidence limits. One can always calculate them
by obtaining the mean and the standard error of the mean from the computer output,
and the appropriate t value from either Table A.3 or the computer output. The rest of
the calculations take very little time to do.
We return now to the example of the 16 gains in weights with x = 311.9 and
s = 142.8 g (see Table 7.1). The standard error of the mean is 142.8/- = 35.7.
The 95% confidence interval is then

&

142.8

311.9 * t[.975]-

m

The number t[.975] is found in Table A.3 to be 2.131 for d.f. = 15. Thus, 97.5% of
the t’s formed from samples of size 16 lie below 2.131, so 95% of them lie between
-2.131 and +2.131. This follows from the same reasoning as that used in discussing
confidence intervals when o was known. The interval is then
142.8
311.9 * 2.131-
m
or
311.9 i 76.1
or
235.8-388.0 g

The interpretation of this confidence interval is as follows: We are “95% confident”
that p lies between 235.8 and 388.08 because, if we keep repeating the experiment
with samples of size 16, always using the formula x 5 t[.975]s/fi for forming a
confidence interval, 95% of the intervals thus formed will succeed in containing p.
In computing a confidence interval using the t distribution, we are assuming that we
have a simple random sample from a normally distributed population. The methods
given in Section 6.4 can be used to decide if the observations are normally distributed
or if transformations should be considered before computing the confidence interval.
Note that in practice we seldom meet all the assumptions precisely; however, the
closer we are to meeting them, the more confidence we have in our results.