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# Download Law of Large Numbers and Central Limit Theorem for Quantitative

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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 p10 and n  1− p 10
```
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