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
σ
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
σ
, 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
μ
has the t distribution
with n1 degrees of freedom
.
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•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).
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 Spring '08
 ABDUS,S.
 Statistics, Normal Distribution, Standard Deviation, Sample standard deviation

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