## Introduction to Sampling

In our daily life we often must examine some given material. We examine fruits before we purchase them, and we… Click here to read more

From basic to higher mathematics

In our daily life we often must examine some given material. We examine fruits before we purchase them, and we… Click here to read more

Population The word population or statistical population is used for all the individuals or objects which we must study. We… Click here to read more

Sample Any part of a population is called a sample. A study of a sample enables us to make some… Click here to read more

Complete Count If we collect information about all individuals in a population, the study is called a complete count or… Click here to read more

If it is not essential to conduct a complete enumeration, then a sample of some suitable size is selected from… Click here to read more

Sometimes, information about each and every unit of the population is required. This is possible only through complete enumeration, because… Click here to read more

Sample Design In sample studies, we have to make a plan regarding the size of the sample, selection of the… Click here to read more

Equal Probability The term equal probability is frequently used in the theory of sampling. This term is quite often not… Click here to read more

The term probability sampling is used when the selection of the sample is purely based on chance. The human mind… Click here to read more

Sampling is called with replacement when a unit selected at random from the population is returned to the population and… Click here to read more

Sampling is called without replacement when a unit is selected at random from the population and it is not returned… Click here to read more

Simple random sample (SRS) is a special case of a random sampling. A sample is called simple random sample if… Click here to read more

Suppose we are interested in the value of a population parameter, the true value of which is $$\theta $$ but… Click here to read more

There are certain sources of errors which occur both in sample surveys as well as in the complete enumeration. These… Click here to read more

Suppose we have a finite population and we draw all possible simple random samples of size $$n$$ without replacement or… Click here to read more

Example: Draw all possible samples of size 2 without replacement from a population consisting of 3, 6, 9, 12, 15…. Click here to read more

From our previous knowledge we are now able to distinguish between a population and a sample, a parameter and statistic,… Click here to read more

In the point estimation procedure we make an attempt to compute a numerical value from sample observations, which could be… Click here to read more

The maximum likelihood estimation method is a very rigorous statistical method of estimation. The word likelihood has the same meaning… Click here to read more

This is probably the most important property that a good estimator should possess. According to this property, if the statistic… Click here to read more

Among a number of estimators of the same class, the estimator having the least variance is called an efficient estimator…. Click here to read more

An estimator $$\widehat \alpha $$ is said to be a consistent estimator of the parameter $$\widehat \alpha $$ if it… Click here to read more

From the discussion on point estimation we know that $$\overline X $$ is the best possible estimator of the population… Click here to read more

From central limit theorem we know that \[Z = \frac{{\overline X – \mu }}{{\frac{\sigma }{{\sqrt n }}}}\] is a standard… Click here to read more

As long as $${\sigma ^2}$$ is known, the confidence interval estimate of a population mean can be obtained by the… Click here to read more

(1) Large Samples The difference between two means is of considerable importance in testing the homogeneity of populations. In this… Click here to read more

The variance of a population can be estimated using the chi-square variate, as explained in previous tutorials. Unlike the $$t$$… Click here to read more

The term standard error has already been introduced very briefly in previous tutorials while discussing the sampling distribution of means…. Click here to read more

The term probable error is seen only in older statistics literature where it used to denote the product of the… Click here to read more

Statistical inference consists of estimating parameters and testing hypotheses. Estimation has already been discussed, so this portion is about testing… Click here to read more

Any opinion or idea may be formed about the population under study. Consider the following statements: The average consumption of… Click here to read more

The hypothesis which is to be tested is called the null hypothesis. It is denoted by $${H_o}$$. It is a… Click here to read more

The hypothesis which is accepted when the null hypothesis has been rejected is called the alternative hypothesis. It is denoted… Click here to read more

The given hypothesis is tested with the help of the sample data. A simple random sample has the full freedom… Click here to read more

A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college… Click here to read more

A statistic is calculated from a sample. To begin with, we assume that the hypothesis about the population parameter is… Click here to read more

All possible values which a test-statistic may assume can be divided into two mutually exclusive groups: one group consisting of… Click here to read more

When the alternative hypothesis $${H_1}$$ is one-sided like $$\theta > {\theta _o}$$ or $$\theta < {\theta _o}$$, then the rejection… Click here to read more

When the rejection region is taken on both ends of the sampling distribution, the test is called a two-sided test… Click here to read more

The null hypothesis $${H_o}$$ is accepted or rejected on the basis of the value of the test-statistic, which is a… Click here to read more

There are 4 people and their heights in inches are 55, 56, 72 and 74. Here height is a characteristic… Click here to read more

Suppose that there are 100 individuals in a certain sample, and the sample size is denoted by $$n$$. These 100… Click here to read more

Tall and short people may further be divided into intelligent and unintelligent people. High intelligence may be denoted by $$B$$… Click here to read more

The classes, $$A,B,AB$$, are called positive classes because they contain all positive attributes. The classes $$\alpha ,\beta ,\alpha \beta $$… Click here to read more

If the class frequencies are observed in a certain sample data and all class frequencies are recorded correctly then there… Click here to read more

Let us consider certain examples before we discuss independence in a formal manner. Example: Consider the following sample data… Click here to read more

By an experimental design, we mean a plan used to collect the data relevant to the problem under study in… Click here to read more

The basic principles of experimental designs are randomization, replication and local control. These principles make a valid test of significance… Click here to read more

A completely randomized (CR) design, which is the simplest type of the basic designs, may be defined as a design… Click here to read more

Statistical data which is recorded with its time of occurrence is called a time series. The yearly output of wheat… Click here to read more