Statistical Parameters , Estimation , Confidence region Submitted By: Pawan Dhamala M.PHARMACY (PHARMACEUTICS)

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Statistical Parameters estimation & Confidence Regions
RR COLLEGE OF PHARMACY
SUBMITTED BY: SUBMITTED TO:
PAWAN DHAMALA PROF. Mr. K MAHALINGAM
2nd SEM , M.PHARMACY
DEPARTMENT OF PHARMACEUTICS

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• Statistical parameter estimation.
• Confidence regions.
• Margin of error
• References.

3. STATISTICAL PARAMETER ESTIMATION
(OR SAMPLE STATISTICS)
• Inferential statistics are used to determine the likelihood that a
conclusion, based on the analysis of the data from a sample, is true and
represents the population studied.
• The two common forms of statistical inference are:
 Estimation
Null hypothesis tests of significance (NHTS).
• There are two forms of estimation:
 Point estimation (maximally likely value for parameter)
Interval estimation (also called confidence interval for parameter)
• Both estimation and NHTS are used to infer parameters. A parameter
is a statistical constant that describes a feature about a phenomena,
population etc.
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4. Examples of statistical parameter include:
• Binomial probability of “success” p (also called “the population proportion”)
• Expected value μ (also called “the population mean”)
• Standard deviation σ (also called the “population standard deviation”)
Point estimates are single points that are used to infer parameters
directly.
• For example,
• Sample proportion pˆ (“p hat”) is the point estimator of p
• Sample mean x (“x bar”) is the point estimator of μ
• Sample standard deviation s is the point estimator of σ
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5. ESTIMATION OF STATISTICAL PARAMETER
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• Estimation of a population parameter is to obtain a guess or an estimate
of the unknown value of the parameter.
• The objective of point estimation is to calculate from the sample
observations, as single number that is likely to be close to the unknown
value of the parameter.
• A statistic intended for estimating a parameter is called a point
estimator. The standard deviation of this estimator is called its standard
error or S E.
• The estimation of the parameters of a statistical model
is one of the fundamental issues in statistics.
• Choosing an appropriate estimator, that is ‘best’ in one
or another respect, is an important task.

6. • Let X1, X2, X3….Xn denote the observations in a random sample of
size n from a population.
• Let µ, population mean and the population standard deviation be
denoted by  respectively.
• The sample mean X￣ is a point estimator of µ. The sample Standard
deviation SD, S is a point estimator .
• The SD of X￣ is called its standard error and is given by /√n.. By
the Central limit theorem, is approximately normal with mean= µ and
SD= /√n.
• Note that the standard Error of X￣ depends on the sample size, n the
larger the sample size, the smaller is the SE, indicating that the
sampling variability will be smaller for larger samples.
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7. MARGIN OF ERROR
• Having computed the point estimator, we now need to compute how accurate this
estimator is. The accuracy of an estimator is measured by a quantity caller its margin of
error or its Error margin.
For a sample mean, the margin of error depends on 3 quantities:
1. The sample size, n
2. The standard deviation 
3. The level of confidence (usually 90%, 95% and 99%)
The level of confidence:
• The level of confidence is a measure of the strength of reliability of the estimator. The
higher the level of confidence, the larger will be the margin of error and the higher will
be our confidence that µ differ from by less than the calculated margin.
• For a sample mean, the error margin EM is calculated as: EM= /√n ×
(Confidence coefficient (Z))
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8. CONFIDENCE REGION/LEVELS
• In statistics, a confidence region is a multi-dimensional generalization
of a confidence interval.
• Confidence regions are multivariate extensions of univariate
confidence intervals.
• Confidence regions usually cover the complete range of data that went
into the model, and incorporate both uncertainty in the parameter
estimates and prediction error.
• Confidence regions are sometimes called inference regions, indicating
that these are regions where one infers something about the likelihood
of the parameters existing.
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9. • For point estimation, we calculate a single number called the point
estimator.
• Instead, it is often more desirable to compute an interval of values that
is likely to contain the true value of the parameter.
• Because the variability of sample to sample, we can never say for sure
if the interval contains the parameter.
• However, we would like to say that the proposed interval will contain
the true value with a specified high probability.
• This probability called the confidence Interval is typically taken as
90%, 95%, 99%. For any confidence level, the corresponding
confidence Interval is computed as:
C I = (mean-EM, mean+ EM). Or P% confidence interval = ± EM
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10. • Computer Applications in Pharmaceutical Research & Development
by Sean Ekins
• www.slideshare.com
• www.google.com
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