1
Sampling Distributions
Chapter 8
8.1 Distribution of the
Sample Mean
A Sampling Distribution of a statistic is a
probability distribution for all possible values of
the statistic computed from a sample of size n.
The sampling distribution of the mean is a
probability distribution of all possible values of
the random variable ‘x bar’ computed from a
sample of size nfrom a population with mean µ
and standard deviation σ.
Example 1: pg. 378- 380
(Pay very close attention!)
Conclusions Based On The
Sampling Distribution of x Bar
1.
The sampling distribution is normally
distributed.
2.
It has mean equal to the mean of the
population.
3.
It has standard deviation less than the
standard deviation of the population.
The Mean and Standard Deviation
of the Sampling Distribution pg.382
Suppose that a simple random sample of size n
is drawn from a large population with mean µ
and standard deviation σ.
The sampling distribution of x bar will have
mean µof x bar = µand standard deviation σof
x bar = σ/ sq. root of n. (see formulas on pg
381)
The standard deviation of the sampling
distribution of x bar, σof x bar, is called the
standard error of the mean.
Example: pg. 389, #12
The Shape of the Sampling
Distribution
If a random variable X is normally
distributed with mean µand standard
deviation σ, then the distribution of the
sample mean, x bar, is normally
distributed with mean µ
xbar
= µand
standard deviation σ
xbar
= σ/√n
The Central Limit Theorem Pg. 385
Suppose a random variable X has
population mean µand standard deviation
σand that a random sample size nis
taken from this population. Then the
sampling distribution of x bar becomes
approximately normal as the sample size n
increases. The mean of the distribution is
µ
xbar
= µand the standard deviation is
σ
xbar
= σ/√n.
