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Statistics Math 121.02 Dr Pendergrass Fall 2007 Law of Large Numbers and Central Limit Theorem for Quantitative Variables Summary Sheet Context: We have a large population, and each individual in the population is associated with a basic measurement, which is quantitative. (Examples: the heights of women in Canada, or the diameters of tubing coming off a major assembly line.) We would like to know what the population mean is, but (as usual) this is not feasible. (By the way: is a parameter or a statisic?) So we decide to estimate from data. We draw an SRS of size n from the population, and then calculate the sample mean x . (By the way, is x random or deterministic?) We'd like to be able to quantify how well x estimates . The Law of Large Numbers. The law of large numbers says that as the sample size n increases without limit, the estimate x gets more and more accurate. Symbolically, we would say x as n ∞ Moreover, the average value of x , denoted by x , is equal to the population average . In symbols, x = The Central Limit Theorem. The central limit theorem says that as the sample size n gets larger and larger, the sampling distribution of x approaches a normal distribution. Moreover, the mean of this distribution is equal to the population mean x = and the standard deviation of the sampling distribution is equal to the standard deviation of the population, divided by the square root of the sample size x = n Statistics Math 121.02 Dr Pendergrass Fall 2007 Law of Large Numbers and Central Limit Theorem for Categorical Variables Summary Sheet Context: We have a large population, and each individual in the population is associated with a categorical variable. (Examples: voters in the USA categorized as either “Democratic”, “Republican”, or “Independent”; or light-bulbs on an assembly line categorized as either “defective” or “not defective”.) We would like to know what proportion p of the entire population falls into a certain category, but (as usual) this is not feasible. (By the way: is p a parameter or a statisic?) So we decide to estimate p from data. We draw an SRS of size n from the population, and then calculate the sample proportion p of individuals in the sample that fall into the given category. (By the way, is p random or deterministic?) We'd like to be able to quantify how well p estimates p. The Law of Large Numbers. The law of large numbers says that as the sample size n increases without limit, the estimate p gets more and more accurate. Symbolically, we would say p p as n∞ Moreover, the average value of p , denoted by p , is equal to the population proportion p. In symbols, p = p The Central Limit Theorem. The central limit theorem says that as the sample size n gets larger and larger, the sampling distribution of p approaches a normal distribution. Moreover, the mean of this distribution is equal to the population proportion p = p and the standard deviation of the sampling distribution is given by p = p 1− p n The normal approximation to the sampling distribution is more accurate for large values of n. For practical purposes, it is safe to use the normal approximation when n and p satisfy both n p10 and n 1− p 10