1
INSE 6220  Week 4
Advanced Statistical Approaches to Quality
•
Inferences about Process Control
•
Sampling and Estimation
•
Confidence intervals
•
Control Charts and hypothesis testing
•
Statistical basis for Control Charts
Dr. A. Ben Hamza
Concordia University
2
Using the normal cdf and pdf
•
We often want to talk about “percentage
points” of the distributionportion in the
tails.
>>
Also, we have:
Example:
/ 2
/ 2
/ 2
/ 2
1
/ 2
(
)
1
(
)
1
(
)
2
(
)
1
2
=
1
2
P Z
z
P Z
z
z
z
z
icdf('normal',1
/2,0,1)
/2
/ 2
(
)
(
)
2
P Z
z
z
0.20/ 2
0.10
0.05/ 2
0.025
1.2816
1.96
z
z
z
z
3
Moments of the population vs. sample statistics
1
2
2
2
2
2
2
1
2
2
2
1
(
)
1
(
)
(
)
1
(
)
(
)
n
X
i
i
n
X
X
X
i
i
E X
X
X
n
Var X
E X
S
S
X
X
n
E X
E X
2
2
2
1
2
2
1
(
,
)
(
)(
)
1
(
)
(
)
( )
(
,
)
(
)
(
)
n
XY
X
y
XY
i
i
i
XY
XY
XY
XY
X
Y
X
Y
S
S
Cov X Y
E X
Y
S
X
X
Y
Y
n
E XY
E X E Y
S
Cov X Y
r
S S
Var X Var Y
•
Mean
•
Variance
•
Standard
Deviation
•
Covariance
•
Correlation
Coefficient
Population
Sample
4
Statistical Inference
•
The purpose of statistical inference
is to obtain information about a population from
information contained in a sample.
•
A population
is the set of all the elements of interest.
•
A sample
is a subset of the population.
•
The sample results provide only estimates
of the values of the population
characteristics.
•
A parameter
is a numerical characteristic of a population.
•
With proper sampling methods
, the sample results will provide “good” estimates of
the population characteristics.
•
In point estimation
we use the data from the sample to compute a value of a
sample statistic that serves as an estimate of a population parameter.
•
We refer to
as the point estimator
of the population mean
.
•
s
is the point estimator
of the population standard deviation
.
•
When the expected value of a point estimator is equal to the population parameter,
the point estimator is said to be unbiased
.
X
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5
Sampling and Estimation
•
Sampling:
act of making observations from populations
•
Random sampling:
when each observation is identically and
independently distributed (IID)
•
Statistic:
a function of sample data; a value that can be computed from
data (contains no unknowns)
–
average, median, standard deviation
•
A
statistic
is a random variable, which itself has a
sampling distribution
–
i.e., if we take multiple random samples, the value for the statistic will be different
for each set of samples, but will be governed by the same sampling distribution
•
If we know the appropriate sampling distribution, we can
reason
about the
population based on the observed value of a statistic
–
E.g. we calculate a sample mean from a random sample; in what range do we
think the actual (population) mean really sits?
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 Fall '10
 Benhamza
 Normal Distribution, UCL

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