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Unformatted text preview: Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw) 81 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 8 Lecture 8 – – Fundamental Sampling Fundamental Sampling Distributions and Data Descriptions Distributions and Data Descriptions Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw) 82 Samples and Populations • P o p u l a t i o n – The totality of observations with which we are conserved – The size of the population may be either finite or infinite – Each observation in a population is a value of a random variable X having some probability distribution f ( x ) • S a m p l i n g – The subset of a population – If the observations of the population are made independently and at random, it is called the random sample. – Otherwise, the sample is said to be biased. Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw) 83 Random Sampling • F o r n random sampling X 1 , X 2 , …, X n from the population f ( x ) , since the identical conditions under which the elements of the sample are selected, it is reasonable to assume that the n random variables X 1 , X 2 , …, X n are independent and each has the identical distribution . • L e t X 1 , X 2 , …, X n be n independent random variables, each having the same probability distribution f ( x ). We then define X 1 , X 2 , …, X n to be a random sample of size n from the population f ( x ) and write its joint probability distribution as ) ( ) ( ) ( ) , , , ( 2 1 2 1 n n x f x f x f x x x f L K = Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw) 84 Some Important Statistics • A n y function of the random variables constituting a random sample is called a statistic . • I f X 1 , X 2 , …, X n represent a random sample of size n , then the sample mean is defined by the statistic • I f X 1 , X 2 , …, X n represent a random sample of size n , then the sample variance is defined by the statistic • T h e sample standard deviation , denoted by S , is the positive square root of the sample variance n X X n i i ∑ = = 1 ( ) 1 1 2 2 − − = ∑ = n X X S n i i ( ) 1 2 1 1 2 2 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ∑ ∑ = = n n X X n S n i i n i i a.k.a Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw) 85 Box and Whisker Plot • This plot encloses the interquartile range of the range of the data in a box that has the median displayed within. – The interquartile range has as its extremes the 75th percentile and the 25th percentile – The whiskers extend the extreme observation in the sample • This plot could show the center of location, variability, and the degree of asymmetry Example : 1.77 Prob. & Stat. Lecture08  sampling distributons and data descriptions (cwliu@twins.ee.nctu.edu.tw)(cwliu@twins....
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This note was uploaded on 08/23/2009 for the course IEE 1036 taught by Professor Cwliu during the Spring '06 term at National Chiao Tung University.
 Spring '06
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