Revised: November 12, 2011
Statistical Inferences from One Sample
(Sections 9.1, 9.2, 10.1, 10.2 and 10.7)
Methods of Statistical Inference:
•
Point Estimation
•
Interval Estimation
•
Significance Tests
Point Estimation
A point estimator is a statistic that specifies how to use the sample data to estimate an
unknown
parameter of a population.
Some commonly used parameters and their estimators:
Parameter
Estimator
μ
= Population mean
X
ˆ
X
μ
=
= Sample Mean
2
σ
= Population variance
2
2
X
X
ˆ
S
σ
=
= Sample variance
p = Population proportion
ˆ
/
p
X
n
=
= Sample proportion
Desirable properties of Point Estimators (
ˆ
θ
):
•
Small variance [
2
ˆ
θ
σ
= Var(
ˆ
θ
)
]
and
small MSE = Var + Bias
2
•
Unbiasedness:
ˆ
( )
E
θ
θ
=
Bias =
ˆ
( )
E
θ
θ

•
Normality
2
ˆ
ˆ
~
( ,
)
N
θ
θ
θ σ
The above estimators have the following properties:
1.
They are
efficient
, i.e. one cannot find other estimators that have smaller standard errors
and these estimators are closer to the true parameter values.
2.
They are
unbiased
. In repeated sampling the estimates average out to give the true values
of the parameters. (S is not an unbiased estimator of
σ
but its bias is small and decreases
as the sample size increases.)
3.
The sample mean and the sample proportion have approximate normal distributions (but
not the sample variance).
General formula for a confidence interval
CI = (Estimate
±
Margin of Error)
ME = Margin of Error = (table value) × (Standard Error of Estimate)
The width of a CI
•
Increases as the confidence level increases
•
Decreases as the sample size increases.
STA3032 Chapters 9 & 10, Page 1 of 14
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Inferences about population Mean, μ
Assumptions that must be satisfied
For making inferences about μ
1.
SRS
2.
Quantitative random variable
3.
Normal population or large n.
Point Estimation:
The sample mean is
a point estimator
of the population mean, μ, i.e.,
X
ˆ
X
μ
=
Interval Estimation:
Confidence interval for the mean of a population, μ
Since we are interested in the population mean, μ, its estimator is the sample mean,
X
, and the
standard error
of the sample mean is
(
)
/
X
X
SE X
n
σ
σ
=
=
.
We want to construct a confidence interval with
level of confidence = 1 – α.
The formula for the margin of error (ME) depends on what is known about the population:
•
IF
the population standard deviation, σ is
KNOWN
and the population has a
normal distribution or n ≥ 30,
THEN
/2
/
X
ME
z
n
α
σ
=
×
.
In this formula z
α/2
is the value of z from the tables of the standard normal distribution that gives
an upper tail probability of α/2, i.e.,
P(Z ≥ z
α/2
) = α/2.
•
IF
the population standard deviation is
UNKNOWN
and the population has a
normal distribution
THEN
(
1,
/2)
/
n
ME
t
S
n
α

=
×
.
In this formula t
α/2
is the value of t from the tables of the tdistribution with df = n – 1, that gives
an upper tail probability of α/2, i.e., P(T
(n1)
≥ t
(n1, α/2)
) = α/2.
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 Summer '08
 Kyung
 Statistics, Normal Distribution, Statistical hypothesis testing

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