mm6 - Inthischapterwelllearnaboutconfidence intervals.

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     In this chapter we’ll learn about ‘confidence  intervals.’  A confidence interval is a range that captures  the ‘true value’ of a statistic with a specified  probability (i.e. ‘confidence’).  Let’s figure out what this means.
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      To do so we need to continue  exploring the principles of statistical  inference: using samples to make  estimates about a population.  See, e.g., King et al.,  Designing  Social Inquiry , on the topic of  inference.
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      Remember that fundamental to statistical  inference are probability principles that allow  us to answer the question: what would  happen if we repeated this random sample  many times, independently and under the  same conditions?
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      According to the laws of probability,  each independent, random sample of  size- from the same population yields  the following:      true value +/- random error
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      The procedure, to repeat, must be a  random sample or a randomized  experiment (or, at very least, independent  observations from a population) in order  for probability to operate.  If not, the use of statistical inference is  invalid.
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     Remember also that sample means are   unbiased estimates of the population mean; & that  the standard deviation of sample means can be  made narrower by (substantially) increasing the  size of random samples- n.   Further: remember that means are less variable  & more normally distributed than individual  observations.
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     If the underlying population distribution is normal,  then the sampling distribution of the mean will also  be normal.  There’s also the Law of Large Numbers.
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     And last but perhaps most important, there’s the  Central Limit Theorem: given a simple random  sample from a population with any distribution of  x when  is large the   sampling distribution of  sample means is approximately normal .
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     That is,  in large samples weighted  averages are distributed as normal  variables .
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      The Central Limit Theorem allows us to use  normal probability calculations to answer  questions about sample means from many  observations even when the population  distribution is not normal  Of course, the sample size must be large  enough to do so.
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     N=30 is a common benchmark threshold for the  Central Limit Theorem, but N=100 or more may  be required, depending on the variability of the  distribution.
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This note was uploaded on 07/11/2011 for the course SYA 6305 taught by Professor Tardanico during the Fall '08 term at FIU.

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mm6 - Inthischapterwelllearnaboutconfidence intervals.

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