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4-4_sampdist - 4.4 Sampling Distribution Models and the...

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4.4 Sampling Distribution Models and the Central Limit Theorem Transition from Data Analysis and Probability to Statistics
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Sampling Distributions Population parameter : a numerical descriptive measure of a population. (for example: μ, σ , p (a population proportion) ; the numerical value of a population parameter is usually not known) Example : μ = mean height of all NCSU students p=proportion of Raleigh residents who favor stricter gun control laws Sample statistic : a numerical descriptive measure calculated from sample data. (e.g, x, s, p (sample proportion))
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Parameters; Statistics In real life parameters of populations are unknown and unknowable . For example, the mean height of US adult (18+) men is unknown and unknowable Rather than investigating the whole population, we take a sample, calculate a statistic related to the parameter of interest, and make an inference. The sampling distribution of the statistic is the tool that tells us how close the value of the statistic is to the unknown value of the parameter.
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DEF: Sampling Distribution The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of values taken by the statistic in all possible samples of size n taken from the same population. Based on all possible samples of size n
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In some cases the sampling distribution can be determined exactly. In other cases it must be approximated by using a computer to draw some of the possible samples of size n and drawing a histogram. Pop size = 5, n = 2, # of poss samples: 5 Pop size: 6; n = 8; # of poss. samples: 6 Pop size: 500,000, n =10; # of samples: 500,000 2 8 10 = = 25 1 679 616 , ,
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If a coin is fair the probability of a head on any toss of the coin is p = 0.5. Imagine tossing this fair coin 5 times and calculating the proportion p of the 5 tosses that result in heads (note that p = x/5, where x is the number of heads in 5 tosses). Objective : determine the sampling distribution of p, the proportion of of heads in 5 tosses of a fair coin. Sampling distribution of p, the sample proportion; an example
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Sampling distribution of p (cont.) Step 1: The possible values of p are 0/5=0, 1/5=.2, 2/5=.4, 3/5=.6, 4/5=.8, 5/5=1 Binomial Probabilities p(x) for n=5, p = 0.5 x p(x) 0 0.03125 1 0.15625 2 0.3125 3 0.3125 4 0.15625 5 0.03125 .03125 .15625 .3125 .3125 .15625 .03125 P(p) 1 .8 .6 .4 .2 0 p The above table is the probability distribution of p, the proportion of heads in 5 tosses of a fair coin.
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Sampling distribution of p (cont.) E(p) =0*.03125+ 0.2*.15625+ 0.4*.3125
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