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Unformatted text preview: Population versus Sample Population – The set of all objects under consideration. Objects in a population are called elements . Examples • Weights of boxes of a particular brand of corn flakes. • The number of minutes to failure of each bulb of all newly made light bulbs in a warehouse at a given time. 1 Another Defintion of Population • All possible outcomes of an experiment can also be values in a population. • Flipping a coin twice. Possible outcomes are: HH, HT, TH, TT. Could make the outcomes numeric using 3, 2, 1, 0 to be the elements corresponding to the above events . • If the experiment is repeated many times, the resulting values constitute a population: 2 , , 3 , 1 , , 2 , . . . 2 Random Sampling • Sample – A subset of the elements from a population. • Simple Random Sample of size n from a finite population is a subset selected in such a way that every subset of n elements is equally likely to be the one selected. (Abbreviation: SRS) • A random sample from a small finite population can be obtained using the Random Numbers (Table 13 of text). • Computer programs are used to draw samples from large populations. 3 Probability, Random Variables, and Probability Distributions • Assume a numeric finite population of N elements, and suppose n 1 of the elements are the number k . • Suppose a SRS of size 1 is taken from this population. • The probability that the number k is selected is n 1 /N . • That is, we define this probability to be the relative frequency of occurrence of k in the population. 4 Probability: Example • Example: N = 10 • Population : 1, 1, 1, 3, 7, 7, 9, 9, 9, 9 • Denote by P ( k ) , the probabilty of the number k being selected • P (1) = 3 / 10 , P (3) = 1 / 10 , P (7) = 2 / 10 , P (9) = 4 / 10 , P (2) = 0 / 10 , P (1 or 3) = 4 / 10 , P ( not 7) = 1 2 / 10 . 5 Random Variables A random variable is a function defined on a numeric population. We will often say that this function value is the value assumed by the random variable . • Example : N = 10 • Population : 1, 1, 1, 3, 7, 7, 9, 9, 9, 9 • Let X be a random variable defined on this population. • X can assume any of the values 1, 3, 7, or 9. 6 Random Variables (continued) • A Random variables are different from ordinary variables because they assume values from the population with associated probabilities. • For e.g., we will say P ( X = 1) = 3 / 10 , P ( X = 7) = 2 / 10 , P ( X = 2) = 0 / 10 , P ( X = 6) = 0 / 10 , P (1 ≤ X ≤ 3) = 4 / 10 , P (1 ≤ X ≤ 9) = 1 . etc. • A probability distribution specifies the probabilities associated with all possible values the random variable X can assume. 7 The M & M Example Example : If X represents the number of RED M&Ms in a vending machine size package, then perhaps the probability distribution of X is something like the following: x 5 6 7 8 9 10 11 P ( X = x ) 2 28 2 28 5 28 7 28 5 28 4 28 3 28 8 The M & M Example (continued) We read this table by finding a value of X, say 8, then reading...
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 Spring '10
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