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Confidence intervals
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Point and interval estimators
There are two kind of estimators:
• Point
• Interval
Point estimator: single statistics used for rstiamte
a paramether of the population. Sample mean is
a point estimator for the mean of the population
, sample variance is a point estimator for the
variance of the population 2, ecc.
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Point and interval estimators
Interval estimator: interval of values that has a
certain probability or confidence to contain the
true value of the paramenter of the population.
The level of confidence is usually (1-)% where 
is the probability that the value stays in the tails of
the distribution, outside the confidence interval.
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Intervallo di confidenza
per la media
noto il valore dello scarto quadratico medio
La statistica per costruire intervalli di confidenza
per la media è
Z
X 

 N (0,1)
n
Ovvero
una
distribuzione
Normale
standardizzata,
“indipendentemente”
dalla
distribuzione originale della variabile X (campioni
sufficientemente grandi). Da tale distribuzione
scaturiscono gli estremi dell’intervallo di
confidenza per la media.
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Confidence interval for the mean
known σ of the poplulation
When the population is normally distributed, the distribution of
the mean is also normal
When population variable X is normally distributed and
is known, a 100(1 − α)% confidence interval for μ is given by
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Confidence interval
Normal curve for Z
with
a
level
of
confidence of 95%
Normal curve for Z
with a level of
confidence of 99%
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Example
•Bolts produced from a firm have an unknown mean diameter, its
variance is 0.01. We keep a sample of n=1000 bolts and we observe
a mean diameter of 1.2 cm. Find the confidence interval at 99% with
a fixed level of confidence of 99%.
•1-α=0.99→ α=0.01 → α/2=0.005 →1-α/2=0.995
•From table of Normal distribution Z(0.995) =2.576
•So the interval is

0.01
0.01 
;1.2  2.576
1.2  2.576
  1.1918;1.2081
1000
1000 

Confidence interval for the mean
unknown σ of the poplulation
Given that σ is not know we need to use its estiamator s.
If we consider the ratio
t
X 
S
n
then the random variable t has the Student’s t distribution with
n − 1 degrees of freedom.
A 100(1−α )% confidence interval for μ is given by
where tα/2 is the upper α/2 point of the Student’s t distribution with
n − 1 degrees of freedom.
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Confidence interval for the mean
unknown σ of the poplulation
If n is very large the t distribution is very close to a Normal distribution.
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Confidence interval for the mean
unknown σ of the poplulation
The table of the Student’s t distribution give the probability (area) on the
right of the indicated value.
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