Observations

  Stating Hypothesis in Testing of Hypothesis Stating Hypothesis, Relation between Interested Conjecture or alternative hypothesis and a Null conjecture or null hypothesis .          Statistics is the science of observations and therefore, one becomes interested to test his build up conjectures about the universe, based on the observations one has collected from the universe through some sampling design. Generally such conjectures are translated into determination of values of some or certain characteristics, also called indicators or variables that are measured on the sampling units. With the help of these indicators, a test statistic is built up to test if the conjecture is supported by the sample (or data). How is it done? We will discuss.             The researcher builds a hypothesis that he wants to get tested. Like suppose he wants to say Regimen A is better than Regimen ...

In Testing of Hypothesis, there is a conjecture, under which the probability distribution function of test-statistics, say T, is known. This conjecture is called null hypothesis and denoted by H0. The test-statistics T is computed based on the observations O = { O1,O2, O3 … On } taken from the universe U = { U1, U2, U3, U4, … UN … }. Here sample size is n, and population size is N or infinity depending upon if population is finite or infinite. For a null hypothesis H0, the test-statistics T represents a summary, for all the observations taken together. This T is generally a scalar function of all the observations. We often take simple null hypothesis. Simple null hypotheses are point hypotheses. Such a null hypothesis makes  test-statistics T to follow only one probability distribution function with a single value of parameter. This single value of parameter is conjectured to be tested. Example H0: μ = μ0 that means null hypothesis is "population mean is μ0." Testing   To tes...

In mathematics, we usually apply deductive rules to prove theorems. The deductive reasoning starts from a general truth (rule) and deduce a specific statement. Since general statement leads to specific statements, there is no chance of having uncertainty in the status of specific statement, because status of specific statement totally depends on the general statements. Examples : (1)- An angle A is acute, if 0 < A < 90; B = 30; then B is acute. (2)- If it rains, then there are clouds in the sky. If we reason that there are no clouds in the sky that means there is no rain. This is reasoning by making first statement into the contra-positive statement. Abstractly if statement A implies B is true then complement of B implies complement of A will also be true. (3)- A implies B, and B implies C, then A implies C. All the above three are deductive reasoning. There is no concept of uncertainty in deductive reasoning because we are going from general to specific, also called top-dow...