7.1
Inference for the Mean of a Population
the sampling distribution of
x
depends on σ
when σ is unknown, we must estimate σ even though we are primarily interested in μ
the sample standard deviation (s) is used to estimate the population standard deviation (σ)
x
has N(μ, σ/
n
) distribution when population has N(μ, σ
)
when σ is unknown, we estimate it with the sample standard deviation (s)
we estimate the standard deviation of
x
by s/
n
Standard error
When the estimated standard deviation is estimated from the data, the result is called the
standard error
of the statistic.
The standard error of the sample mean is
SE
x
=
s/
n
onesample z statistic (6.2) =
(
x
μ
0
) / (σ/
n
)
basis for inference about μ when σ is known
x
distributed normally (or approximately normally) used Table A
when we substitute s/
n
for
σ/
n
, our statistic is not distributed normally
the statistic now has a t distribution
The t distributions
Suppose that an SRS of size n is drawn from a N(μ, σ) population.
Then the onesample t
statistic
t = (
x
μ
0
) / (s/
n
)
has the t distribution
with n1 degrees of freedom
.
•there is a different t distribution for each sample size (Table D)
•a particular t distribution is specified by giving the degrees of freedom
•we use t(k) to stand for the t distribution with k degrees of freedom
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View Full DocumentWhy degrees of freedom = n1?
Recall CH 1.
If we know
x
and the values of n1 of the
observations, we can figure out the value of the last observation and, hence the standard
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 Fall '08
 ABDUS,S.
 Normal Distribution, Standard Deviation

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