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Chapter 4 Evaluating Analytical Data 77

4
EXAMPLE .14
What is the 95% confidence interval for the analgesic tablets described in
Example 4.13, if an analysis of five tablets yields a mean of 245 mg of aspirin?
SOLUTION
In this case the confidence interval is given as

)(
.
( 196 7)
6
m=245 ± =245 mg ±mg
5
Thus, there is a 95% probability that the population’s mean is between 239 and
251 mg of aspirin. As expected, the confidence interval based on the mean of
five members of the population is smaller than that based on a single member.

4 4 Probability Distributions for Samples
D.
In Section 4D.2 we introduced two probability distributions commonly encoun-
tered when studying populations. The construction of confidence intervals for a
normally distributed population was the subject of Section 4D.3. We have yet to ad-
dress, however, how we can identify the probability distribution for a given popula-
tion. In Examples 4.11–4.14 we assumed that the amount of aspirin in analgesic
tablets is normally distributed. We are justified in asking how this can be deter-
mined without analyzing every member of the population. When we cannot study
the whole population, or when we cannot predict the mathematical form of a popu-
lation’s probability distribution, we must deduce the distribution from a limited
sampling of its members.

Sample Distributions and the Central Limit Theorem Let’s return to the problem
of determining a penny’s mass to explore the relationship between a population’s
distribution and the distribution of samples drawn from that population. The data
shown in Tables 4.1 and 4.10 are insufficient for our purpose because they are not
large enough to give a useful picture of their respective probability distributions. A
better picture of the probability distribution requires a larger sample, such as that
–
2
shown in Table 4.12, for which X is 3.095 and s is 0.0012.
The data in Table 4.12 are best displayed as a histogram, in which the fre- histogram
quency of occurrence for equal intervals of data is plotted versus the midpoint of A plot showing the number of times an
each interval. Table 4.13 and Figure 4.8 show a frequency table and histogram for observation occurs as a function of the
range of observed values.
the data in Table 4.12. Note that the histogram was constructed such that the mean
value for the data set is centered within its interval. In addition, a normal distribu-
–
2
2
tion curve using X and s to estimate mand s is superimposed on the histogram.
It is noteworthy that the histogram in Figure 4.8 approximates the normal dis-
tribution curve. Although the histogram for the mass of pennies is not perfectly
symmetrical, it is roughly symmetrical about the interval containing the greatest
number of pennies. In addition, we know from Table 4.11 that 68.26%, 95.44%,
and 99.73% of the members of a normally distributed population are within, re-
spectively, ±1s, ±2s,and ±3s. If we assume that the mean value, 3.095 g, and the
2
sample variance, 0.0012, are good approximations for mand s , we find that 73%,