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Unformatted text preview: STAT 200 Chapter 5 Sampling Distributions The binomial distribution for counts (Section 5.1) 1. The Binomial Experiment (a) The experiment consists of n identical trials. The number of trials is fixed in advance. (b) The outcomes on each trial can be dichotomized into two complementary cate- gories: success and failure. (c) The probability of the success event, p , is the same from trial to trial. The probability of the failure event is 1- p . (d) The n trials are independent of each other. In other words, the results of previous trials do not affect the result of the current trial. Under the conditions of a binomial experiment, the random variable X representing the number/count of successes out of the n trials is a binomial random variable with parameters n and p . We write X Bin ( n,p ). 2. For X Bin ( n,p ), the probability that X will take on a value of k is given by P ( X = k ) = n k ! p k (1- p ) n- k , k = 0 , 1 , 2 , ,n where n k ! = n ! k !( n- k )! is called the binomial coefficient (Another notation is n C k ). It gives the total number of ways (or combinations of successes and failures) of having X = k . Note that k ! = k ( k- 1) ( k- 2) ... 2 1 and 1! = 1, 0! = 1. 3. Mean (expected value), Variance and SD for a Binomial Random Variable X Mean X : E ( X ) = n X k =0 k P ( X = k ) = n X k =0 k n k ! p k (1- p ) n- k = np 1 Variance 2 X : V ( X ) = n X k =0 ( k- np ) 2 P ( X = k ) = n X k =0 ( k- np ) 2 n k ! p k (1- p ) n- k = np (1- p ) Standard deviation X = SD ( X ) = q np (1- p ) Interpretation of E ( X ): the average number of successes if you are to repeat the binomial experiment (each with n trials) many times. Interpretation of V ( X ): a measure of the variability of the numbers of successes if the binomial experiment (each with n trials) is repeated many times. Sampling distribution (Section 3.3) Population vs. sample, parameter vs. statistic The complete collection of individuals that we want to study about is called the population . A census provides a means to obtain complete and accurate informa- tion about a population of interest. However, it can be very costly and time inefficient to carry out. Sometimes a census is impossible. It is more practical to study a sample , i.e., a subset of individuals selected from a population. A sample can provide reliable information about a population as long as the sample is representative of the population. A systematically nonrepresentative sample is said to be biased . By randomly selecting individuals from the population, the sample produced tend to have characteristics comparable to the population. The chance of obtaining a non- representative sample is thus minimized....
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This note was uploaded on 01/17/2011 for the course STAT 200 taught by Professor Karim during the Spring '08 term at The University of British Columbia.
- Spring '08