LECTURE 3: STATISTICAL INFERENCE
Point Estimate:
The estimator (e.g.
X
) applied to the sample gives you a
single
guess
(point estimate) about the unknown parameter (e.g. the population mean
μ
).
Interval Estimate of Confidence Interval
: It gives a
range of values
likely to
contain the unknown parameter (e.g.
μ
) at a certain
confidence level
(i.e. with
certain, usually high, probability).
What does this mean?
Suppose we repeatedly drew samples from a distribution with true population
mean
μ
. For each of these samples, suppose we constructed a 95% confidence
interval for
μ
. In 95% of these samples, the calculated confidence interval around
the sample mean would include the true population mean.
Next, we calculate
two-sided confidence intervals
for a population mean. We can
do the same for any other unknown parameter of a distribution, provided that we
know the (fixed sample or asymptotic) distribution of an estimator that consistently
estimates the unknown parameter of interest.
CONFIDENCE INTERALS FOR THE MEAN
Case 1: Normal random sample – known variance:
X
i
~ i.i.d.N(
μ
,
σ
2
) where
μ
unknown,
σ
2
known.
•
We specify a
confidence level
, (1 –
α
) × 100%, where
α
is a small number
between 0 and 1, e.g.
α
= 0.01, 0.05, or 0.10, which corresponds to
confidence levels of 99%, 95%, and 90%.
•
α
is the
significance level.
•
It is natural to use
X
as our estimator of
μ
. Here we know
)
n
σ
,
μ
(
N
~
X
2
Standardizing:

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*