Statistical Inference: An Overview
Statistical inference refers to the process of drawing conclusions about a
population based on a sample of data. Since it's often impractical to study an
entire population, statisticians use samples to make educated guesses, or
inferences, about larger populations. This approach relies on probability
theory to quantify the uncertainty and variability present in the data.
These problems formulated mathemtically by assuming the observed data to be
values of a random variable/ vector $X$. The probability distribution of $X$
is modelled using the standard techniques of fitting distribution.
The problem of statistical inference deals with this situation that is to come
up on the basis of the data call it $X_{1},X_{2},X_{3},........X_{n}$ about
suitable statements assertion about the unknown parameter $\theta$,
An exact estimate is called the problem of point estimation and
identifying an interval is called the problem of
interval estimation.
Some information regarding the unknown feature of the population maybe
available to the experiment. He/she good like to check whether the information
is teenager in the light of the random sample drawn from the population this
is called the 'testing of hypothesis'.
Terms in Inference
Population:
A statistical population is an aggregate universe of numerical/ qualitative,
records, of the measurements on certain characteristics of interest.
For example: the salaries of employees in an industrial sector, the weight at
birth of children in geographical region etc.
Parameter
The specific characteristics of the population such as average (mean, median)
variability (standard deviation, range), the maximum, minimum etc in which the
experiment is interested are called parameters. Usually te parametric
inference assumes a distribution $f(x,\theta)$ where $\theta$ is the parameter
characterizing population.
Statistic
A statistic is a function of the sample observations. For example let
$X_{1},X_{2},X_{3},........X_{n}$ denote a random sample from a sample.
A function $T(X)=(X_{1},X_{2},X_{3},........X_{n})$ is called a statistic.
Estimator
Any function of the random sample which is used to estimate the unknown value
of the given parametric function $g(\theta)$ is called an estimator.
If $X = (X_{1},X_{2},X_{3},........X_{n})$ is a random sample from a
population with the probability distribution $P_{\theta}$, a function $d(X)$
used for estimating $g(\theta)$ is known as estimator. Let $x =
(x_{1},x_{2},x_{3},........x_{n})$ be a realisation of $X$ . Then $d(X)$
is called an estimate.
Parameter Space
It is the set of all possible values of a parameter.
`\theta \rightarrow \Theta`
There are two primary types of statistical inference:
Estimation –
This involves estimating population parameters, such as the mean or
proportion, using sample statistics. Estimation can be further divided into:
Point Estimation:
Provides a single value as an estimate.
Interval Estimation: Gives a range of values (confidence intervals) within
which the parameter is likely to fall.
Hypothesis Testing – This method tests assumptions (hypotheses) about a
population parameter. Based on sample data, statisticians decide whether to
reject or fail to reject a hypothesis
Applications of Statistical Inference
Statistical inference is used in various fields, including:
-
Quality Control: Determining if a production process meets certain standards
based on a sample.
-
Medicine: Testing the effectiveness of new treatments through clinical
trials.
-
Economics: Estimating key indicators like unemployment or inflation rates
from surveys.
-
Environmental Science: Predicting climate changes based on historical data.
- Importance of Statistical Inference
-
By using statistical inference, researchers and professionals can make
informed decisions and predictions without needing data from the entire
population. This method not only saves time and resources but also provides
a structured way to manage uncertainty and variability in real-world
scenarios.
