5.2 - Samplingdistributionofsamplemean We take many random...

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Sampling distribution of sample mean We take many random samples of a given size n from a population with mean μ and standard deviation σ. Some sample means will be above the population mean and some will be below, making up the sampling distribution. Sampling distribution of sample mean Histogram of some sample averages
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Because the probability of drawing one individual at random depends on the frequency of this type of individual in the population, the probability is also the shaded area under the curve. The shaded area under a density curve shows the proportion , or %, of individuals in a population with values of X between x 1 and x 2 . % individuals with X such that x 1 < X < x 2 Random variable and population distribution
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A general fact Let X 1 , X 2 , …., X n be a simple random sample with replacement from a population. Then X 1 , X 2 , …., X n are independent and identically distributed random variables. Their common probability distribution is the population distribution. If X 1 , X 2 , …., X n are a simple random sample without replacement from a population. Then X 1 , X 2 , …., X n are identically distributed random variables with the population distribution as their common probability distribution. These random variables are not independent, but the dependence is negligible if we only sample a small fraction of the population.
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From now on we say that X 1 , X 2 , …., X n are a random sample if they are independent and identically distributed. 1. Simple random sample from a population or 2. Independent repeated measurements of the same quantity.
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Suppose the population contains three different  numbers 1, 2, and 3 with the same proportion Draw n numbers randomly from the population with replacement . Let the observations be X 1 , X 2 , …., X n . Then X 1 , X 2 , …., X n are random variables, with possible values 1, 2 and 3. Each of X 1 , X 2 , …., X n has the same probability distribution p (X i = 1) = p (X i = 2) = p (X i = 3) = 1/3. This common distribution is the population distribution . Also, X 1 , X 2 , …., X n are mutually independent.
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The mean of the population distribution is equal to 1(1/3) + 2(1/3) + 3(1/3) = 2, (population mean μ) and its variance is equal to 1 2 (1/3) + 2 2 (1/3) + 3 2 (1/3) –(2) 2 = 2/3 (population variance σ 2 )
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This note was uploaded on 09/04/2010 for the course STAT 131 taught by Professor Isber during the Spring '08 term at University of California, Berkeley.

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5.2 - Samplingdistributionofsamplemean We take many random...

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