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Inferential Statistics
Dr. Dennis S. Mapa
Prof. Manuel Leonard F. Albis
UP School of Statistics
Decision Making Under Uncertainty

A credit investigator recommends favorable action
to an application for credit using certain
characteristics of the client (credit scoring).

Using regression analysis, a financial analyst
concludes that a financial ratio is significantly
correlated with a set of business variables.
Decision Making Under Uncertainty

An economist concludes that a set of
variables can be considered as “deep”
determinants of economic growth using the
results of his/her econometric model from a
sample of cross-country data.
Decision Making Under Uncertainty
These are examples of decision making that
employ statistical methods.
The statistical methods used are under the
area of statistical inference or inferential
statistics.
What is Inferential Statistics?
Statistical inference refers to methods by which
one uses sample information to make inferences or
generalizations about a population.
Deals with inferences about some characteristic(s)
of the population.
Recall the other area of statistics is descriptive
statistics which is composed of methods concerned
with collecting, describing and analyzing a set of
data without drawing inferences on the population
to which the data belong.
Point Estimation
An estimator is any statistic whose value is used to
estimate an unknown parameter. A realized value
of an estimator is called an estimate.
Two Important Statistics
the mean
the standard deviation
The Arithmetic Mean
the most common “average”
the population mean is denoted by the greek
letter µ (“mu”),
N

 Xi
i 1
N
the sample mean (estimator) is denoted by
n
X
 Xi
i 1
n
Standard Deviation

the positive square root of the variance
Population Standard
Deviation
N


Sample Standard
Deviation
X i   
2
i 1
N
 X i  X 
n
s
i 1
2
n 1
The standard deviation is known as volatility.
Proportions
 In a binomial experiment where the outcomes can be
considered as “success” or “failure” (example: having a
positive or negative return), the parameter of interest is
the proportion (p) of sucess.
 A success is usually denoted by 1 and failure 0.
 A (point) estimator of the proportion (p) is,
X
pˆ 
n
where X represents the number of successes in a
sample of n observations
Statistical Inference
 Statistical inference refers to methods by which
one uses sample information to make
inferences or generalizations about a
population.
The two areas of statistical inference are
interval estimation and hypothesis testing.
Statistical Inference
 Confidence Interval Estimation uses sample data to
calculate the lower and upper bound of an interval such
that there is a specified probability that the true
parameter value lies within the interval. This is in
contrast to finding a single value in point estimation.
 Hypothesis Testing is the process of making decisions on
whether or not the sample agrees with the researcher’s
assertion regarding some characteristic of the
population.
Sampling Distributions
An important concept in statistical inference is the
notion of sampling distributions.
The probability distribution function of a statistic
is called its sampling distribution.
Recall that statistics (such as the sample mean)
are used to estimate the unknown parameters of
the population (such as the population mean)
Sampling Distributions
 A statistic is a random variable whose value depends
only on the observed sample and may vary from sample
to sample.
 The sampling distribution of a statistic will depend on
the size of the population, the size of the sample, and
the method of choosing the sample.
 The standard deviation of the sampling distribution is
called the standard error of the statistic. It tells us the
extent to which we expect the values of the statistic to
vary from different possible samples.
What is the Standard Error?
It is a measure of error of estimation.
It is the positive square root of the variance
of the estimator.
It measures, on the average, the dispersion
of each possible value of the estimator from
the actual value being estimated.