Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 2.13 Summary 79 obtain the remaining numbers in the sample. If we decide to proceed down the page, the next number (immediately below the 5) is a 2. So our second sampled person would be number 2. Proceeding, we next come to an 8, but there are only seven elements in the population. Thus, the 8 is ignored, and we continue down the column. Two more 5s then appear, but they must both be ignored because person 5 has already been selected. (The chip numbered 5 has been removed from the pile.) Finally, we come to a 1, and our sample of three is completed with persons numbered 5, 2, and 1. Any starting point can be used in a random number table, and we may proceed in any direction from the starting point. However, if more than one sample is to be used in any problem, each should have a unique starting point. In many situations the population is conceptual, as in an observation made during a laboratory experiment. Here the population is envisioned to be the infinitely many measurements that would be obtained if the experiment were to be repeated over and over again. If we wish a sample of n = 10 measurements from this population, we repeat the experiment ten times and hope that the results represent, to a reasonable degree of approximation, a random sample. Although the primary purpose of this discussion was to clarify the meaning of a random sample, we would like to mention that some sampling techniques are only partially random. For instance, if we wish to determine the voting preference of the nation in a presidential election, we would not likely choose a random sample from the population of voters. By pure chance, all the voters appearing in the sample might be drawn from a single city—say, San Francisco—which might not be at all representative of the population of all voters in the United States. We would prefer a random selection of voters from smaller political districts, perhaps states, allotting a specified number to each state. The information from the randomly selected subsamples drawn from the respective states would be combined to form a prediction concerning the entire population of voters in the country. In general, we want to select a sample so as to obtain a specified quantity of information at minimum cost. 2.13 Summary This chapter has been concerned with providing a model for the repetition of an experiment and, consequently, a model for the population frequency distributions of Chapter 1. The acquisition of a probability distribution is the first step in forming a theory to model reality and to develop the machinery for making inferences. An experiment was defined as the process of making an observation. The concepts of an event, a simple event, the sample space, and the probability axioms have provided a probabilistic model for calculating the probability of an event. Numerical events and the definition of a random variable were introduced in Section 2.11. Inherent in the model is the sample-point approach for calculating the probability of an event (Section 2.5). Counting rules useful in applying the sample-point method were discussed in Section 2.6. The concept of conditional probability, the operations of set algebra, and the laws of probability set the stage for the event-composition method for calculating the probability of an event (Section 2.9). Of what value is the theory of probability? It provides the theory and the tools for calculating the probabilities of numerical events and hence the probability

www.downloadslide.com 80 Chapter 2 Probability distributions for the random variables that will be discussed in Chapter 3. The numerical events of interest to us appear in a sample, and we will wish to calculate the probability of an observed sample to make an inference about the target population. Probability provides both the foundation and the tools for statistical inference, the objective of statistics. References and Further Readings Cramer, H. 1973. The Elements of Probability Theory and Some of Its Applications, 2d ed. Huntington, N.Y.: Krieger. Feller, W. 1968. An Introduction to Probability Theory and Its Applications, 3d ed., vol. 1. New York: Wiley. ———. 1971. An Introduction to Probability Theory and Its Applications, 2d ed., vol. 2. New York: Wiley. Meyer, P. L. 1970. Introductory Probability and Statistical Applications, 2ded. Reading, Mass.: Addison-Wesley. Parzen, E. 1992. Modern Probability Theory and Its Applications. New York: Wiley-Interscience. Riordan, J. 2002. Introduction to Combinatorial Analysis. Mineola, N.Y.: Dover Publications. Supplementary Exercises 2.143 Show that Theorem 2.7 holds for conditional probabilities. That is, if P(B) > 0, then P(A|B) = 1 − P(A|B). 2.144 Let S contain four sample points, E 1 , E 2 , E 3 , and E 4 . a List all possible events in S (include the null event). b In Exercise 2.68(d), you showed that ∑ n ( n ) i=1 i = 2 n . Use this result to give the total number of events in S. c Let A and B be the events {E 1 , E 2 , E 3 } and {E 2 , E 4 }, respectively. Give the sample points in the following events: A ∪ B, A ∩ B, A ∩ B, and A ∪ B. 2.145 A patient receiving a yearly physical examination must have 18 checks or tests performed. The sequence in which the tests are conducted is important because the time lost between tests will vary depending on the sequence. If an efficiency expert were to study the sequences to find the one that required the minimum length of time, how many sequences would be included in her study if all possible sequences were admissible? 2.146 Five cards are drawn from a standard 52-card playing deck. What is the probability that all 5 cards will be of the same suit? 2.147 Refer to Exercise 2.146. A gambler has been dealt five cards: two aces, one king, one five, and one 9. He discards the 5 and the 9 and is dealt two more cards. What is the probability that he ends up with a full house?