Download Law of Large Numbers and Central Limit Theorem for Quantitative

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