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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,
n
σ
) 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
n
s
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
x
SE
=
n
s
onesample
z
statistic (6.2) =
z
=

n
x
μ
)
(
0
basis for inference about μ when σ is known
x
distributed normally (or approximately normally) used Table A
when we substitute
n
s
for
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
=

n
s
x
)
(
0
has the
t
distribution
with
n
1 degrees of freedom
.
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t
statistic
t
=

n
s
x
)
(
0
μ
has the
t
distribution
with
n
1 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
•The density curves of the
t
(
k
) distributions are similar in shape to the standard normal
curve (unimodal, symmetric about 0, and bellshaped).
•As the degrees of freedom
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This note was uploaded on 02/23/2011 for the course STAT 2700 taught by Professor Bill during the Spring '11 term at Adelphi.
 Spring '11
 Bill
 Standard Deviation

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