Page 236 - Applied Statistics And Probability For Engineers

c06.qxd 5/14/02 9:54 M Page 197 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

6-3 STEM-AND-LEAF DIAGRAMS 197

f 1x 2, f 1x 2, p , f 1x 2, respectively, and by independence the joint probability density function
1
n
2
of the random sample is f 1x , x , p , x 2 f 1x 2 f 1x 2 p f 1x 2.
X 1 X 2 p X n 1 2 n 1 2 n
Definition
, X , p , X are a random sample of size n if (a) the X ’s are
The random variables X 1 2 n i
independent random variables, and (b) every X has the same probability distribution.
i

To illustrate this definition, suppose that we are investigating the effective service life of
an electronic component used in a cardiac pacemaker and that component life is normally dis-
in
tributed. Then we would expect each of the observations on component life X 1 , X 2 , p , X n
a random sample of n components to be independent random variables with exactly the same
normal distribution. After the data are collected, the numerical values of the observed life-
times are denoted as x 1 , x 2 , p , x n .
The primary purpose in taking a random sample is to obtain information about the unknown
population parameters. Suppose, for example, that we wish to reach a conclusion about the pro-
portion of people in the United States who prefer a particular brand of soft drink. Let p represent
the unknown value of this proportion. It is impractical to question every individual in the popula-
tion to determine the true value of p. In order to make an inference regarding the true proportion
p, a more reasonable procedure would be to select a random sample (of an appropriate size) and
use the observed proportion ˆp of people in this sample favoring the brand of soft drink.
The sample proportion, ˆp is computed by dividing the number of individuals in the sam-
ple who prefer the brand of soft drink by the total sample size n. Thus, ˆp is a function of the
observed values in the random sample. Since many random samples are possible from a pop-
ulation, the value of ˆp will vary from sample to sample. That is, ˆp is a random variable. Such
a random variable is called a statistic.

Definition
A statistic is any function of the observations in a random sample.

We have encountered statistics before. For example, if X, X , p , X n is a random sample of
2
size n, the sample mean X, the sample variance S 2 , and the sample standard deviation S are
statistics.
Although numerical summary statistics are very useful, graphical displays of sample data
are a very powerful and extremely useful way to visually examine the data. We now present a few
of the techniques that are most relevant to engineering applications of probability and statistics.

6-3 STEM-AND-LEAF DIAGRAMS

The dot diagram is a useful data display for small samples, up to (say) about 20 observations.
However, when the number of observations is moderately large, other graphical displays may
be more useful.
For example, consider the data in Table 6-2. These data are the compressive strengths in
pounds per square inch (psi) of 80 specimens of a new aluminum-lithium alloy undergoing eval-
uation as a possible material for aircraft structural elements. The data were recorded in the order
of testing, and in this format they do not convey much information about compressive strength.
Questions such as “What percent of the specimens fail below 120 psi?” are not easy to answer.