CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTION

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Population

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Sample (x1, x2, x3, , xn) x, sample average s, sample standard deviation Histogram

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Figure 6-3 Relationship between a population and a sample.

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chassis structural element to be normally distributed with mean and variance 2 . We could refer to this as a normal population or a normally distributed population. In most situations, it is impossible or impractical to observe the entire population. For example, we could not test the tensile strength of all the chassis structural elements because it would be too time consuming and expensive. Furthermore, some (perhaps many) of these structural elements do not yet exist at the time a decision is to be made, so to a large extent, we must view the population as conceptual. Therefore, we depend on a subset of observations from the population to help make decisions about the population. De nition A sample is a subset of observations selected from a population.

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For statistical methods to be valid, the sample must be representative of the population. It is often tempting to select the observations that are most convenient as the sample or to exercise judgment in sample selection. These procedures can frequently introduce bias into the sample, and as a result the parameter of interest will be consistently underestimated (or overestimated) by such a sample. Furthermore, the behavior of a judgment sample cannot be statistically described. To avoid these dif culties, it is desirable to select a random sample as the result of some chance mechanism. Consequently, the selection of a sample is a random experiment and each observation in the sample is the observed value of a random variable. The observations in the population determine the probability distribution of the random variable. To de ne a random sample, let X be a random variable that represents the result of one selection of an observation from the population. Let f(x) denote the probability density function of X. Suppose that each observation in the sample is obtained independently, under unchanging conditions. That is, the observations for the sample are obtained by observing X independently under unchanging conditions, say, n times. Let Xi denote the random variable that represents the ith replicate. Then, X1, X2, p , Xn is a random sample and the numerical values obtained are denoted as x1, x2, p , xn. The random variables in a random sample are independent with the same probability distribution f(x) because of the identical conditions under which each observation is obtained. That is, the marginal probability density function of X1, X2, p , Xn is

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6-3 STEM-AND-LEAF DIAGRAMS

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f 1x1 2, f 1x2 2, p , f 1xn 2, respectively, and by independence the joint probability density function of the random sample is fX1 X2 p Xn 1x1, x2, p , xn 2 f 1x1 2 f 1x2 2 p f 1xn 2. De nition The random variables X1, X2, p , Xn are a random sample of size n if (a) the Xi s are independent random variables, and (b) every Xi has the same probability distribution.

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To illustrate this de nition, suppose that we are investigating the effective service life of an electronic component used in a cardiac pacemaker and that component life is normally distributed. Then we would expect each of the observations on component life X1, X2, p , Xn in 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 lifetimes are denoted as x1, x2, p , xn. 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 proportion 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 population 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 p of people in this sample favoring the brand of soft drink. The sample proportion, p 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, p is a function of the observed values in the random sample. Since many random samples are possible from a population, the value of p will vary from sample to sample. That is, p is a random variable. Such a random variable is called a statistic. De nition A statistic is any function of the observations in a random sample.

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We have encountered statistics before. For example, if X, X2, p , Xn is a random sample of 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.

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