Transcript Chapter 10 – Sampling Distributions

Chapter 10 – Sampling
Distributions
Math 22
Introductory Statistics
The Sampling Distribution of a
Sample Statistic
 The sampling distribution of a
sample statistic is the distribution of
values for a sample statistic obtained
from repeated samples.
The Sampling Distribution of
the Sample Mean
Let x be the mean of a sample of size
n from any population that has mean 
and standard deviation .
For all sample sizes n, the sampling
distribution of the sample mean:
 Is exactly normally distributed.
The Sampling Distribution of
the Sample Mean
 Is centered at
, the mean of the
population.
 Has a standard deviation of  / n ,
where  is the standard deviation of
the population.
 Note: We don’t know what the true
population mean and population
standard deviation are.
Central Limit Theorem (CLT)
 The sampling distribution of sample
means will become normal as the
sample size increases.
The Probability of the Sample
Mean
 Old z score
z
x  x
x

x

 Note: This z score is used to calculate
the probability of a single observation.
The Probability of the Sample
Mean
 New z score
z
x  x
x
x

/ n
 Note: This z score is used in
calculating the probability of the sample
mean.
The Sampling Distribution of
the Sample Proportion
CLT Applied to the Sample Proportion.
If the sample size n, is sufficiently large
(both n and n(1   ) are at least 5), then
the sampling distribution of the sample
proportion:
 Is approximately normally distributed
 Is centered at  , the true proportion of
success in the Bernoulli population.
 (1   )
 Has a standard deviation of
n