CHAPTER 11 –SAMPLING DISTRIBUTIONS
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Any topic listed on this document and not covered in class must be studied “On Your Own” (OYO)
READ INTRODUCTION
Section 11.1 – PARAMETERS AND STATISTICS (p. 292) – Listen to STATS TUTOR
 Population
o Parameter – fixed, unknown number that describes the population
o Population mean – notation (μ)
o Population proportion – notation (p)
 Sample
o Statistic – known value calculated from a sample
o Sample mean – notation (x̅)
o Sample proportion – notation (p̂)
 Distinguishing parameters and statistics
Example: A properly chosen sample of 1600 people across the United States was asked if they regularly
watch a certain television program, and 24% said yes. The parameter of interest here is the true proportion
of all people in the U.S. who watch the program, while the statistic is the value 24% obtained from the
sample of 1600 people.
Section 11.2 – STATISTICAL ESTIMATION AND THE LAW OF LARGE NUMBERS (p. 293) - Listen to STATS TUTOR
 Statistical inference, what does it mean?
 X-bar as a random variable
 Law of large numbers:
Consider sampling at random from a population with true mean µ.
As the number of (independent) observations sampled increases, x̅ gets closer to μ.
Section 11.3 – SAMPLING DISTRIBUTION (pg. 296) - Listen to STATS TUTOR
 Distribution of sample means for samples of size n
o Simulation
 Population distributions
 Sampling distributions of a statistic (in this case the sample mean)
Section 11.4 – THE SAMPLING DISTRIBUTION OF X-BAR (pg. 299) - Listen to STATS TUTOR
 Distribution of sample means for samples of size n
o Center
o Spread
 X-bar is an unbiased estimator of mu
 Shape of the distribution of sample means
 If individual observations have the N (  ,  ) distribution, then the sample mean x-bar of an SRS of size n has
the N (  ,  / n )
Section 11.5 – THE CENTRAL LIMIT THEOREM (pg. 301) - Listen to STATS TUTOR
Draw an SRS of size n from any population with mean  and standard deviation  . The Central Limit Theorem says
that when n is large, the sampling distribution of the sample means x-bar is approximately normal with
x  
and
x 

n
x 


The standard deviation of x-bar is called the Standard Error of the mean


If x is normally distributed, so is the x-bar distribution, regardless of sample size
If the sample size is large (n  30) , the x-bar distribution is approximately normally distributed, regardless of the
distribution of x.
n
Summary

If x ~ (  ,  ) and n is large, then x ~ N (  ,  / n ) regardless of the shape of the distribution of x.

How large a sample size n is needed for the x distribution to be close to Normal depends on the population
distribution; if x is too far from normal, more observations are required

If x is normal, then x is also normal regardless of the sample size n
Formula for z-score used when finding probabilities in the distribution of x-bars
z = (score – mean) / (std dev of distribution)
z = (score – mean) / (standard error)
z
(x  )
(

n
)

A parameter in a statistical problem is a number that describes a population, such as the population mean σ
To estimate an unknown parameter, use a statistic calculated from a sample, such as the sample mean

The law of large numbers states that the actually observed mean outcome
the population as the number of observations increases.

The population distribution of a variable describes the values of the variable for all individuals in a
population.

The sampling distribution of a statistic describes the values of the statistic in all possible samples of the
same size from the same population.

When the sample is an SRS from the population, the mean of the sampling distribution of the sample
mean is the same as the population mean . That is, μ is an unbiased estimator of μ.

The standard deviation of the sampling distribution of is σ/
foran SRS of size n if the population has
standard deviation σ. That is, averages are less variable than individual observations.
must approach the mean μ of

When the sample is an SRS from a population that has a Normal distribution, the sample mean
Normal distribution.

Choose an SRS of size n from any population with mean μ and finite standard deviation σ. The central limit
theorem states that when n is large the sampling distribution of is approximately Normal. That is, averages
are more Normal than individual observations. We can use the N(μ, σ/
distribution
alsohas a