Sampling

Central Limit Theorem

General Idea: Regardless of the population distribution model, as the sample size increases, the sample mean tends to be normally distributed around the population mean, and its standard deviation shrinks as n increases. Certain conditions must be met to use the CLT.

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The samples must be independent The sample size must be “big enough”

CLT Conditions

Independent Samples Test

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“Randomization”: Each sample should represent a random sample from the population, or at least follow the population distribution. “10% Rule”: The sample size must not be bigger than 10% of the entire population. Large Enough Sample Size Sample size n should be large enough so that np≥10 and nq≥10

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Example: Is CLT appropriate?

It is believed that nearsightedness affects about 8% of all children. 194 incoming children have their eyesight tested. Can the CLT be used in this situation?

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Randomization: We have to assume there isn't some factor in the region that makes it more likely these kids have vision problems. 10% Rule: The population is “all children” - this is in the millions. 194 is less than 10% of the population. np=194*.08=15.52, nq=194*.92=176.48 We have to make one assumption when using the CLT in this situation.

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Central Limit Theorem (Sample Mean)

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X1, X2, ..., Xn are n random variables that are independent and identically distributed with mean μ and standard deviation σ. X = (X1+X2+...+Xn)/n is the sample mean We can show E(X)=μ and SD(X)=σ/√n CLT states: X −μ   N 0,1 σ / n as n→∞

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Implication of CLT

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 We have: X −μ  N 0,1 σ / n

  N  μ , σ 2 / n Which means X

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So the sample mean can be approximated with a normal random variable with mean μ and standard deviation σ√n.

Proportions of a Sample

Let's say we have a population with probability p of a certain characteristic (and q=1-p). We have a random sample of n from the population. What is the mean and standard...