Sampling Distributions. Suppose we use the sample mean, y, as our “best guess”
of the population mean, μ. We assume it will be somewhat close to the target, ...
Sampling Distributions Suppose we use the sample mean, y, as our “best guess” of the population mean, . We assume it will be somewhat close to the target, but not exactly on every time. It is also intuitive that if we were to take another sample from the same population and calculate the sample mean for that sample, it would not only be off the target, but also be slightly different from the first sample mean. Definition The variability among random samples from the same population is called sampling variability. Definition A probability distribution that characterizes some aspect of sampling variability is called a sampling distribution. A sampling distribution tells us how close the resemblance between the sample and population is likely to be. You can imagine describing the sampling distribution of a statistic by repeatedly taking samples from the same population over and over again, computing the statistic, and plotting all the statistics on a probability histogram. This distribution would have a shape, a mean, standard deviation, etc… We’ll study the sampling distribution of the sample mean, y. Sampling Distribution of y Given a random sample, Y1, Y2, … , Yn where E[Yi] = and SD(Yi) = , we have the following facts about the distribution of y. Mean μY = E[y] = σ Standard Deviation σy = SD(y) = √n
Shape If Yi has a normal distribution, then so does y. Central Limit Theorem (CLT) As the sample size grows large, the distribution of y becomes approximately normal.
The CLT is app proximatelyy true for ffinite n and d the apprroximation n improves as n gets llarger. metimes we e only need a few ob bservationss for it to kkick in, som metimes we need mo ore. This Som rd depends on ho ow “far fro om normall” the data is to beginn with. Fro om 3 ed. of text…
Exam mple 5.13 from 3rd ed. Y denottes the num mber of eyye facets in n a fruit flyy. Clearly, Y is not conttinuous an nd hence, ccannot be normally d distributedd. The follo owing figurre illustrates how the CLT still kicks in.
m example e 5.2.4 Lett Y denote the weight of seeds,, with Y ~ N N(=500, = 120). From Find d P(y > 550 0) for a random samp ple with n = = 25.
Chap pter 5
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