Basic concepts in Testing of Hypothesis

Tests of Hypothesis

Sampling theory ©

             Sampling is a part of our day to day life. Sampling is preferred to complete enumeration due to the fact that it is less time consuming, less expensive more accurate and reliable. Also in some cases sampling is the only possible method of data collection.

Population

            Population is a group of items, units or subjects which is under reference of study. It may consist of finite of infinite number of units (Universe).

Sample

         Sample is a part or fraction of a population selected on some basis. It consists of a few items of population. In principle, a sample should be such that it is a true representative of the population. Usually a random sample is selected. By random sampling, we mean the sampling in which each and every unit of the population has an equal and independent chance of being included in the sample. 

       In order to draw inference about certain phenomenon, sampling is a well accepted tool. Entire population can not be studied due to several reasons (stated above). In such a situation sampling is used. A properly drawn sample is much useful in drawing reliable conclusions. Here we draw a sample from probability distribution rather than a group of objects.

Random sample from a continuous probability distribution

              A random sample from a  continuous probability distribution f(x,ϴ) is nothing but the values of independent and identically distributed ( i. i. d.) random variables with the common probability distribution f(x,ϴ).

Definition:  If x1, x2,----, xn  are i. i. d. random variables with p. d. f. f(x, ϴ), then they from a random sample  from the population with p. d. f. f(x,ϴ) .

Note: i) For drawing inference we use the numerical values of x1, x2, ----, xn.                               ii) The joint p. d. f. of x1, x2,----, xn  is

Statistic

              Using the random sample x1, x2,----, xn  we draw conclusion about the unknown probability distribution. However probability distribution can be studied if the parameter ϴ is known. We use sample observations for this purpose. The sampled observations are summarized. The summarized quantity is called as statistic (estimator).

Definition:  If x1, x2,----, xn  is a  random sample from a  probability distribution  f(x,ϴ), then t = t t(x1, x2,----, xn ) a function of sample values which does not involve the unknown parameter ϴ is called as a  statistic( estimator).

Parameter

               A function based on population values is called as parameter. If f(x,ϴ) is a p. d. f. then the constant ϴ, involved in it is called as parameter. Statistic is a random variable and parameter is a constant. Since statistic is a random variable, it possesses some probability distribution; it may not be the same as that of the parent distribution f(x,ϴ). However parameter being a constant does not possess probability distribution.

Estimator  

               An estimator is a function of sample values for estimating the population parameter. A particular value of an estimator from a fixed set of values of a random sample is known as estimate. An estimate stands for the value of a parameter. For ex:  sample mean X-bar  is an estimate of population mean  µ  and sample variance S2 is an estimate of population variance ơ2

.

Unbiased Estimate

                                  A statistic or estimator t is said to be an unbiased estimate of population parameter ϴ,  if     E(t) = ϴ,  i.e. E( statistic or estimator ) = parameter.

Sampling distribution of statistic and Standard error

                   If x1, x2,----, xn  is a  random sample from a  probability distribution  f(x,ϴ), then the probability distribution of T = t (x1, x2,----, xn ) is called as its sampling distribution and standard deviation of t is called as its standard error ( S.E.).

                      
In testing of hypothesis, standard error of T is important.  Some typical statistics along with standard error are :  

where p1 and p2 are proportions obtained using two samples from two populations with proportions P1 and P2 .  ©

***

 

Comments

Post a Comment

Popular posts from this blog

Unit 1 : Multiple Regression , Multiple Correlation and Partial Correlation 1.1: Multiple Linear Regression (for trivariate data)

Image
  Unit 2 : Multiple Regression , Multiple Correlation and Partial Correlation  Unit 2.1 Multiple Linear Regression  (for tri variate data)      Some statistical methods serve as forecasting or estimation techniques. One of such is regression analysis. We have learnt concept of linear regression and correlation with respect to two variables. It is called as simple correlation and simple regression.      In practice, the variable under study is influenced by two or more variables. For example: national income based on several variables such as agricultural yield, industrial production, import, export production of minerals, marine wealth etc.       For accuracy in prediction it is inevitable to include all interrelated  variables in the regression model. Among such variables, whose value is to be predicted  is called as dependent variable or  response variable and remaining variables are treated as independen...
Read more
Image
  RELIABILITY THEORY INTRODUCTION:  ©                In day-to-day life we come across several types of machines, equipment, systems such as radio set, T. V. set, vehicle, electric bulb, computer, music system etc. We always ask questions to manufacture that how long the machine will work without any trouble. What is the guarantee period of the item? Answers to such questions are based on the life time of the components of machine as well as the entire assembly. A reliability theory deals with extensive study of life time of components and machines. This is one of the important branch having wide applications in the manufacturing process.                    Reliability theory is concerned with determining a probability that, a system of several components is functioning i.e. working at a specified time ‘t’. The se...
Read more

B.Sc. I (Sem. II Paper III) Unit 2, 2.1 INDEX NUMBERS : निर्देशांक , देशनांक , सूचकांक

Image
  Unit ... 2 The time has come, the Walrus said  To speak of many things,  Of shoes, and ships, and sealing wax,  Of cabbages and kings                                                                       -  l ewis Carroll 2.1  Index Numbers introduction:           Index Number is a tool used to measure the changes in prices of commodities, industrial and agricultural production, sales, imports, exports, etc. It was first developed by an Italian economist Mr. Carli in 1764 for comparison of prices of commodities. The fields in which the index numbers are used are economics, trade, stock market, Government Organizations etc. The popularly used index numbers are- i) Bombay Stock Exchange (BSE) SENSEX Index Number. ii) NSE ind...
Read more