STAT METH RESEARCH 1
STAT METH RESEARCH 1 STA 6166
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This 10 page Class Notes was uploaded by Golden Bernhard on Friday September 18, 2015. The Class Notes belongs to STA 6166 at University of Florida taught by Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/206575/sta-6166-university-of-florida in Statistics at University of Florida.
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Date Created: 09/18/15
Sampling Distributions You have seen probability distributions of various types The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as weights heights etc The binomial distribution is an example of a discrete distribution because the possible outcomes are only a discrete set of values 0 l n The value of the binomial random variable is the number of successes out of a random sample of 11 trials in which the probability of success on a particular trail is 7 The Binomial Distribution as 21 Sampling Distributions The nature of the binomial distribution makes it a sampling distribution In other words the value of the binomial random variable is derived from a sample of size n from a population You can think of the population as being a set containing 0 s and 1 s with a 1 representing a success and 0 representing a failure The term success does not necessarily imply something good nor does failure imply something bad The sample is obtained as a sequence of 11 trials with each trial being a draw from the population of 0 s and 1 s Such trials are called Bernoulli trials On the first trial you draw a value from 0 1 with P17r Then you draw again in the same way and do this repeatedly a total of n times So your sample will be a set such as 1 1 0 1 0 1 in the case for n6 This set has 4 1 s and 2 0 5 so the value of the binomial random variable is y4 The value of the binomial distribution is the sum of the outcomes from the trials that is the number of 1 s The Binomial Distribution as 21 Sampling Distribution Recall that the binomial probability formula is nl 397iy l 7239 quot y P02 successes in 11 trials yKn y Where 7 probability of success on single trial and n number or trials Mean of the binomial distribution quot7 Variance of the binomial distribution 71750 7r This probability is derived from the sampling distribution of the number of successes that would result from a very large number of samples The Binomial distribution as 21 Sampling Distribution Consider a population that consists 60 of 1 s and 40 of 0 s If you draw a value at random you get a 1 with probability 6 and a 0 with probability 4 Such a draw would constitute a Bernoulli trial with Pl6 Suppose you draw a sample of size n and add up the value you obtain This would give you a binomial random variable Now suppose you do this again and again for a very large number of samples The conceptual results make up a conceptual population Here are the histograms that correspond to the population for values ofn equal to l 2 3 6 10 and 20 Notice how the shapes of the histograms change as 11 increases The distributions become more moundshaped and symmetric Relative Frequency O Mmbo lm Histotram of B1 6 Distribution Successes Relative Frequency Histogram of B2 6 Distribution 0 1 Successes Relative Frequency Histogram of B3 6 Distribution Successes Histogram of B6 6 Distribution 0 0 Relative Frequency 0 1 2 3 4 5 6 Successes Histogram of B10 6 Distribution gt U S w 3 U39 2 LL w 2 2 w a Successes Histogram of Successes gt U S w 3 U39 2 LL w 2 g m a 012 3 4 5 6 7 8 91011121314151617181920 Successes The Binomial Sampling Distribution and Statistical Inference Here are the values of Pyk and Pylt k for n20 and k617 Pyklt005 for nlt6 or ngt17 6 7 8 9 10 11 12 13 14 15 16 17 P046 005 015 035 071 117 160 180 165 124 075 035 012 P023 k 006 021 057 128 245 404 584 750 874 949 984 996 Remember that these are the probabilities for 7r6 Suppose you are sampling from a distribution with unknown 7 If you drew sample with y12 would you have reason to doubt that 7r6 Note yn6 If you drew sample with y10 would you have reason to doubt that 7r6 Note yn5 If you drew sample with y6 would you have reason to doubt that 7r6 Note yn3 The Binomial Sampling Distribution and Statistical Inference Here are the numeric values of Pyk for n10 3 decimal places 0 1 2 3 4 5 6 7 8 9 10 P0 k 000 002 011 042 111 201 251 215 121 040 006 P023 k 000 002 012 055 166 367 618 833 954 994 100 Remember that these are the probabilities for 7r6 Suppose you are sampling from a distribution with unknown 7 If you drew sample with y6 would you have reason to doubt that 716 Note yn6 If you drew sample with y5 would you have reason to doubt that 716 Note yn5 If you drew sample with y3 would you have reason to doubt that 716 Note yn3 Normal Distributions and Sampling Distributions One of the most use il statistics is the sample mean 7 Statistical inference based on 7 is derived from its sampling distribution If the population from Which the sample was obtained is norm distributed With mean p and variance 62 then the sampling distribution of y is normal Moreover the mean of the sampling as mean p and variance 6 n If y NNUwZ i1n then yNNoml n ELFvw39s39m39m39u39ma39aLFv39wm Normal Distributions and Sampling Distributions If the distribution from which the samples were obtained is not normal then the sampling distribution of y is only approximately normal but the distribution becomes more nearly normal as 11 increases This is the Central limit Theorem Central Limit Theorem If y is a random variable with mean u and variance 02 then the sampling distribution of y is approximately normal with mean u and variance 0211
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