Transcript Sampling Distribution

Population and Sample
The entire group of individuals that we want information about
is called population.
A sample is a part of the population that we actually examine in
order to gather information.
Moore & McCabe (1999, p.256)
Parameters and Statistics
A parameter is a number that describes the population. A
parameter is a fixed number, but in practice we do not know its
value.
A statistic is a number that describes a sample. The value of a
statistic is known when we have taken a sample, but it can change
from sample to sample. We often use a statistic to estimate an
unknown parameter.
Moore & McCabe (1999, p.268)
Population
Sample
Statistic:
Parameter:
Proportion
Proportion p
Count
Count
Mean 
Median
Mean
x
Median
p̂
Sampling Distribution
The sampling distribution of a statistic is the distribution of
values taken by the statistic in all possible samples of the same
size from the same population.
Moore & McCabe (1999, p.269)
Sampling Distribution of Proportions
Sampling Distribution of Means
p
proportion  p̂
p̂
p̂
Sample
Proportion = p
Population
p̂
Central Limit Theorem for a Sample Proportion
Suppose that a simple random sample of size n is taken from a
large population in which the true proportion possessing the
attribute of interest is p. Then the sampling distribution of the
sample proportion p̂ is approximately normal with mean p and
standard deviation p1  p  .
n
This approximation becomes more and more accurate as the
sample size n increases, and it is generally considered to be
valid, provided that np  10 and n1  p   10 .
Rossman et al. (2001, p.375)

mean  x
mean  x
mean  x
mean  x
Sample
Mean = 
Population
Central Limit Theorem for a Sample Mean
Suppose that a simple random sample of size n is taken from a
large population in which the variable of interest has mean 
and standard deviation . Then, provided that n is large (at least
30 as a rule of thumb), the sampling distribution of the sample
mean x is approximately normal with mean x and standard
deviation  .
n
The approximation holds with large sample sizes regardless of
the shape of the population distribution. The accuracy of the
approximation increases as the sample size increases. For
populations that are themselves normally distributed, the result
holds not approximately but exactly.
Rossman et al. (2001, p.398)