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# M312_1 - Probability and Statistics II M-312 Lecture 1 1...

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Probability and Statistics II M-312 Lecture 1 1. Random Samples 1.1. Statistics and Their Distributions The observations in a single sample are denoted by n x x x , , , 2 1 . Before we obtain data there is uncertainty about the values of each x i . Because of this uncertainty, before the data becomes available we view each observation as a random variable and denote the sample by n X X X , , , 2 1 . This variation in observed values in turn implies that the value of any function of the sample observations – such as sample mean, sample standard deviation, etc. – also varies from sample to sample. That is, prior to obtaining n x x x , , , 2 1 there is uncertainty as to the value of x , the value of s , and so on. Definition 1.1. A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. Therefore, a statistic is a random variable and will be denoted by an uppercase letter; a lowercase letter is used to represent the calculated or observed value of the statistic. Definition 1.2. The rv’s n X X X , , , 2 1 are said to form a (simple) random sample of size n if 1. The X i ’s are independent rv’s. 2. Every X i has the same probability distribution. Conditions 1 and 2 can be paraphrased by saying that the X i ’s are independent and identically distributed (iid). If sampling is either with replacement or from an infinite (conceptually) population, Conditions 1 and 2 are satisfied exactly. These conditions will be approximately satisfied if sampling is without replacement, yet the sample size n is much smaller than the population size N . In practice, if 05 . 0 N n (at most 5% of the population is sampled), we can proceed as if the X i ’s form a random sample. There are two general methods for obtaining information about a statistic’s sampling distribution. One method involves calculations based on probability rules, and the other involves carrying out a simulation experiment.

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DERIVING A SAMPLING DISTRIBUTION Example 1.1. A large automobile service center charges \$40, \$45, \$50 for a tune-up of four-, six-, and eight-cylinder cars, respectively. If 20% of its tune-ups are done on four- cylinder cars, 30% on six-cylinder cars, and 50% on eight-cylinder cars, then the probability distribution of revenue from a single randomly selected tune-up is given by x 40 45 50 p ( x ) 0.2 0.3 0.5 (1.1) with 25 . 15 5 . 46 5 . 0 50 3 . 0 45 2 . 0 40 ) ( 5 . 46 5 . 0 50 3 . 0 45 2 . 0 40 ) ( 2 2 2 2 2 3 1 2 2 3 1 = - + + = - = = + + = = = = μ σ i i i i i i x p x x p x Suppose on a particular day only two servicing jobs involve tune-ups. Let X 1 = the revenue from the first tune-up, and X 2 = the revenue from the second. Suppose that
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M312_1 - Probability and Statistics II M-312 Lecture 1 1...

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