ANOVA:The acronymANOVArefers toanalysis of varianceand is a statistical procedure used to test the degree to which two or more groups vary or differ in an experiment. In most experiments, a great deal of variance (or difference) usually indicates that there was a significant finding from the research. Analysis of variance ( abbreviated as ANOVA) is an extremely useful technique concerning researches in the field of economics, biology, education, psychology, sociology, business/industry and in research of several other disciplines. This technique is used when multiple sample cases are involved. As we know the significance of the difference between the means of two samples can be judged through either z-test or the t-test, but difficulty arises when we happen to examine the significance of the difference amongst more than two sample mean at the same time. This technique is important in all t those situations where we want to compare more than two populations such as in comparing the yield of crop from several varieties of seeds, the gasoline mileage of four automobiles etc.Example1. (Exercise)Example2.In the majority of experiments, you need first anull hypothesisand analternative hypothesis. The null hypothesis is the assumption that there will be no differences between groups that are tested and therefore no significant results will be revealed. The alternative hypothesis, on the other hand, is the hypothesis stating that there will be a difference between groups as indicated by the ANOVA performed on the data that is collected.Let's use an experiment scenario to help explain things. Imagine that you are running an experiment to see if there is a relationship between people's religion and what they consider the ideal family size to be. You would likely do this by recruiting individuals from different religious groups and asking them to report what they consider the ideal amount of children in a family should be. Let us further say that you ended up recruiting 10 Catholics, 10 Protestants, and 10 Jewish individuals to answer this question.In this case, you have oneindependent variable, which isreligion, that is thought to have an effect on the opinion of ideal family size, which is thedependent variablein this scenario. Religion should affect the ideal family size andsince it is the factor thought to influence the difference. Additionally, this experiment includes three different levels of the independent variable. In this case, the three levels are the three different groups of religions in which one is Catholic, one is Protestant, and one is Jewish.The fact that we have differing levels of the independent variable of religion is what allows us to carry out an ANOVA. Let's say that after asking all the people in all three groups what they consider the ideal number of children in a family to be, you record each person's answer and then calculate themean, or average, number reported by each collective group. You discover that the average number of children reported by the Catholic group is 3, for the Protestant group it is 2, and for the Jewish group it is 1.At first glance, it may seem like there is a definite difference between these three groups in their opinion on the ideal number of children. However, we must keep in mind that this could be due to chance and these numbers could be very different if we asked 10 different Catholics, 10 different Protestants, and 10 different Jewish individuals. Therefore, an ANOVA is a good test to use as it will control for this and determine if there really is a difference between the three groups beyond mere random chance.In this particular example, the difference between the average of the three groups were statistically significant (as computed by ANOVA test) and not due to chance. This means that religious affiliation does influence opinions on the ideal number of children in a family. Therefore we shown that the null hypothesis is false since there is a significant difference between the three religious groups and the alternative hypothesis has been proven true.