eChapters

In factorial experiment it may sometime happen that certain factors require bigger plots than other for convenience of operation of the experiment.  For example, in an agricultural experiment involving two factors, say, irrigation and fertilizer, it is organizationally very inconvenient to apply different levels of irrigation to small neighbouring plots.  But there is no such difficulty for the application of different levels of fertilizer. To meet such situations it is desirable to have different sizes of the experimental units in the same experiment.

In the foregoing experiments performed either in CRD or RBD or LSD. We were primarily concerned with the comparison and estimation of the effects of a single set of treatments like varieties of paddy, different methods of cultivation etc., and such experiments which deal with one factor only may be called simple experiments. In factorial experiment the effects of several factors of variation are studied and investigated simultaneously in a single experiment, such experiments are known as factorial experiments. In these experiments an attempt is made to estimate the effects of each factors and also the interaction effects i.e. the variation in the effect of one factor as a result to different levels of other factors.  These experiments provide an opportunity to study not only the individual effects of each factor but also interactions.  In many biological and clinical trials factors are likely to have interaction. Therefore, factorial experiments are more informative in such investigations.

In many situations, experiments may not known in advance which contrasts they wish to compare, or they may be interested in more than (m – 1) possible comparisons.  In many exploratory experiments, the comparisons of interest are fixed only after preliminary examinations of the data.
 
13.1 Scheffe’s Method for Comparing All Contrasts

Scheffe (1953) has proposed a method for comparing any all possible contrasts between treatment means.  In the Scheffe method, the type I error is at most a for any of the possible comparisons.

12.1 Introduction
 
The design of the experiment includes (i) planning of the experiment (ii) obtaining the relevant information from it regarding the statistical hypothesis under study and (iii) making a statistical analysis of the data. Design of experiment may be defined as the logical construction of the experiment in which the degree of the uncertainty with which the inference is drawn may be well defined. It may also be redefined as the choice of treatments, the method of assigning treatments to experimental units and arrangement of experimental units in varies patterns to suit the requirement of particular problems, are combined known as the design of experiment.

11.1 Introduction

The analysis of variance is a powerful statistical tool for tests of significance.  The test of significance based on t-distribution is an adequate procedure only for testing the significance of the difference between two sample means. In a situation when we have three or more samples to consider at a time an alternative procedure is needed for testing the hypothesis that all the samples are drawn from the same population, i.e., they have the same mean. For example five fertilizers are applied of four plots each of wheat and yield of wheat on each of the plot is given we may be interested in finding out whether the effect of these fertilizers on the yields is significantly different or in other words, whether the samples have come from the same normal population. The answer to this problem is provided by the technique of analysis of variance. Thus basic purpose of the analysis of variance is to test the homogeneity of several means.

10.1 Introduction

In previous lessons we have been dealing with problems connected with variation occurring in any one variable, say the income of a person, the height or weight of an individual, the wage of a worker in a factory, the sales of a shop, demand of a commodity etc.  The values of a single variable were obtained from every member of the population and the different characteristics were described by certain descriptive constants after grouping into frequency distribution.  In this lesson, we shall be dealing with problems wherein there are paired variables and every individual or member of the population exhibits two values, one for each of the variables under consideration.  Example of the two variables which are to be studies together are (i) the income and expenditure of a household, (ii) the price and demand or supply of a commodity, (iii) the height and weight of a student, and (iv) the grain yield and straw yield of paddy, (v) Height of father and height of son, etc.

We have seen the procedures of testing hypothesis about population parameters in chapter 11. Frequently our interest is to test whether a set of observed values are in agreement with those which would occur if some specified hypothesis were true. The X² statistic provides a measure of agreement between such observed and expected frequencies.

The X² distribution has a number of applications in testing of hypotheses. It is used to test, (1) the independence of attributes, (2) the  goodness of fit, (3) the homogeneity of variances, (4) the homogeneity of correlation coefficients, and (5) the linkage in genetic problems. For applying X² test procedure we have to use only the actual observed frequencies and not the percentages or ratios. Further, the samples should be independent and the observations within a sample should be non-overlapping. The expected frequency in each category should be more than 5. The number of observations should be large, say, more than 50.

8.1 Statistical Hypotheses

in attempting to reach decisions about the populations, it is useful to make assumptions or guesses about the populations involved. When the assumption or statement about a phenomenon occurring under certain conditions is formulated as proposition it is called a scientific hypotesis. We can construct criteria by which a scientific hypothesis may be ‘the  yield of a new paddy variety will be 3500 kg per hectare’. In statistical language it may be stated as ‘the random variable (yield of paddy) is distributed normally with mean 3500 kg/ha’. Such a statement or proposition about the probability distribution of a random variable is called the statistical hypothesis.

7.1 Meaning of Sampling
    
Before we consider sampling theory, we first define the terms population and sample.
 
The word ‘population’ or ‘universe’ in statistics is used to refer to any collection of individuals or of their attributes or of results of operations which can be numerically specified. For example the assemblage of heights of all the individuals in the population is called a population of heights.  Similarly we may consider the population of weights, wages, yields of grain; prices of wheat, mileages of automobile tyres, etc. a population containing a finite number of individuals or members is called a finite population.  For instance the population f ages of the workers in a factory constitute a finite population of ages. A population with infinite number of members is known as infinite population.  For example the population of pressures of temperatures at various points in the atmosphere is an infinite population. 

It has been a general notion that if an experiment is conducted under identical conditions, values so obtained would be similar. But the experiences of people have dispelled this belief. Observations are always taken about a factor or character under study, which can take different values, and the factor or character is termed as variable.
Hence, we have a set of outcomes (sample points) of a random experiment. A rule that assigns a real number to each outcome (sample points) is called random variable. The rule is nothing about a function of the variable, say, X, that assigns a unique value to each sample point of the sample space.

In day-to-day life, one comes across many statements which create doubt about certain happenings and one may want to determine the chance of occurrence of a particular happening, e.g., if one desires to know the chance of rain based on some metrological observations.  In the case of coin tossing, one wants to know what will be the chance of winning if a person always stakes for head in a fixed number of tossing. There can be innumerable such queries which one would like to make and meanwhile evaluate mathematically the chance of the occurrence of such possibilities.  This invoked the need of probability theory. The theory of probability came in existence when problems of games were referred to mathematicians like Karl Pearson, Laplace and others. Complicated applications involving advanced theory of probability and axionmatic approach of probability are not dealt within this book.

4.1 Introduction and Measuring

A measure of central tendency exhibits only one of the important characteristics of distribution, namely they typical representative figure to the whole set of its values. This measure alone will not adequately describe a set of observation unless all the observations are alike. Moreover one cannot understand completely about the data set with a measure of central tendency.  Because in real practical situation there may exits a number of sets of observations (or different distributions) whose measures of central tendency are same but they may differ individually from each other in a number of ways.  In particular two sets of values have the same mean but whilst one set of data is grouped close to the mean the other one is more spread out.  The following examples will illustrate this view point clearly.

3.1 Introduction

One of the most important objectives of statistical analysis is to summaries the information contained in the data which may be considered either as a population or as a sample (or sub-population). If the set of data consists of all conceivably possible (or hypothetically possible) observations of a certain phenomenon or a variable, we call it as population. If a set of data contains only a part of these observations, that is selected values of a variable, we call it as sample. Frequency distribution and graphical representation summarize to some extent.  They reduce a mass of data to a more understandable form but not to a single value. The aim of this lesson is to determine descriptive measures or statistical constants which will describe the nature of the data.

2.1 Quantitative Classification

It the data are classified on the basis of phenomenon which is capable of quantitative measurement like age, height, weight, prices, production, income expenditure, sales, profits, etc., it is termed as quantitative classification. The quantitative phenomenon under study is known as variable.  For example the daily incomes of different retail shops in a town may be classified as under.

1.1 Definition of Statistics

Statistics has been defined differently by different ‘Statisticians’ from time to time. These definitions emphasize precisely the meaning, scope and limitations of the subject.  The reasons for such a variety definitions may be stated as follows:

    i.    The field of utility of statistics has been increasing steadily and
    ii.    The word statistics has been used to give different meanings in singular (the science of statistical methods) and plural (numerical set of data) senses.

In order to have a clear grasp of the subject, we give below some selected definitions of statistics which are grouped under two main heads: ‘Statistics as numerical data’ and ‘Statistics as Methods’.  The former one is in plural sense and the latter singular sense.

9.1  Write True/False

Q.1    In the subject matter of order statistic, the useful beta distribution is written as :

8.1 Write True/False

Q.1    Any linear function of sample observations which is used to estimate the population parameter or parametric function,is called the “Linear estimator.”

7.1  Write True/False

Q.1    All the problem of statistical inference may be considered as the problem of “Decision making in the face of uncertainty.”
Q.2    The problem of point estimation is the problem if deciding which value in the “parameter space” Wq is the true value of the parameter.

6.1  Write True/False

Q.1    A procedure to make an inference about the unknown values of parameters of one or more than one distribution in which the sample size say n is a ramdom variable, is known as “Sequential procedure.”
Q.2    The principal feature of sequential procedure is a ‘Sampling Scheme’ which provides a rule under which one decides at each stage of sampling whether to stop or continue the sampling work.

5.1  Write True/False

Q.1    The null hypothesis H₀ is said to be composite if it does not specify the joint probability of x₁, x₂, ......, x_n completely.
Q.2    We may write null hypothesis as :
    H₀ : θΣ Ωθ_0-