Applied Statistics and Probability for Engineers

6-3 STEM-AND-LEAF DIAGRAMS 197

We have encountered statistics before. For example, if is a random sample of

size n, the sample mean the sample variance , and the sample standard deviationSare

statistics.

Although numerical summary statistics are very useful, graphical displaysof 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.

X, S^2

X, X 2 ,p, Xn

respectively, and by independence the joint probability density function

of the random sample is fX 1 X 2 pXn 1 x 1 , x 2 ,p, xn 2 f 1 x 12 f 1 x 22 pf 1 xn 2.

f 1 x 12 , f 1 x 22 ,p, f 1 xn 2 ,

The random variables are a random sample of size nif (a) the Xi’s are

independent random variables, and (b) every Xihas the same probability distribution.

X 1 , X 2 ,p, Xn

Definition

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-

tributed. Then we would expect each of the observations on component life in

a random sample of ncomponents 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

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 prepresent

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 pˆof people in this sample favoring the brand of soft drink.

The sample proportion, pˆis 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, pˆis a function of the

observed values in the random sample. Since many random samples are possible from a pop-

ulation, the value ofpˆwill vary from sample to sample. That is, pˆis a random variable. Such

a random variable is called a statistic.

x 1 , x 2 ,p, xn.

X 1 , X 2 ,p, Xn

A statisticis any function of the observations in a random sample.

Definition

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