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UASTAT141Ch18 - Ch 18 Sampling Distributions Expanded defn...

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Ch. 18 - Sampling Distributions Expanded def’n: A parameter is: - a numerical value describing some aspect of a pop’n - usually regarded as constant - usually unknown A statistic is: - a numerical value describing some aspect of a sample - regarded as random before sample is selected - observed after sample is selected The observed value depends on the particular sample selected from the population; typically, it varies from sample to sample. This variability is called sampling variability . The distribution of all the values of a statistic is called its sampling distribution . Def’n: p ˆ = proportion of ppl with a specific characteristic in a random sample of size n p = population proportion of ppl with a specific characteristic The standard deviation of a sampling distribution is called a standard error . General Properties of the Sampling Distribution of p ˆ : Let p ˆ and p be as above. Also, p ˆ µ and p ˆ σ are the mean and standard deviation for the distribution of p ˆ . Then the following rules hold: Rule 1 : p ˆ µ = p . (Textbook uses ) ˆ ( p µ ) Rule 2 : n pq n p p p = = ) 1 ( ˆ σ . (standard error Æ p ˆ ˆ σ ) e.g. Suppose the population proportion is 0.5. a) What is the standard deviation of p ˆ for a sample size of 4? 25 . 0 4 ) 5 . 0 1 ( 5 . 0 ) 1 ( ˆ = = = n p p p σ b) How large must n (sample size) be so that the sample proportion has a standard deviation of at most 0.125? p p p n ˆ ) 1 ( σ = Æ 16 125 . 0 ) 5 . 0 1 ( 5 . 0 ) 1 ( 2 2 ˆ = = = p p p n σ Rule 3 : When n is large and p is not too near 0 or 1, the sampling distribution of p ˆ is approximately normal. The farther from p = 0.5, the larger n
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