Need for Randomization

We may recall that in Chapter 7, discussing the experiment on the benefit of a hypothetical plant food, and also in other con­texts, we mentioned the word “random” quite a few times. We may now ask, Why random? Why not pick the first forty or so plants of one kind that the experimenter came

Applications of Randomization

Obtaining samples or specimens from a group (also known as a lot or population) of any kind is an activity wherein the applica­tion of randomness is of utmost significance. In the case of the study by pairing, a toss of the coin serves the purpose. Random­ization has applications beyond sampling, for example, in the

Methods of Randomization

Throwing the coin and wagering heads or tails is the simplest random process. If there are more than two possibilities, obvi­ously, throwing the coin is useless. Then, we may resort to the method of picking paper strips with hidden numbers, this being suitable for any number including two. In the hope of reducing the

Meaning of Randomization

After dealing with these various contexts—there are many more—of randomization, we may ask, What is really meant by randomization? The closest we can get to the meaning of “ran­dom” is “done without previous calculation” or “left to chance.” Any attempt to define randomization is known to be philosophi­cally hopeless for it begs the idea

Replication

We have already used this word, even before defining it, as we shall do now. Replication is repetition with some difference. When we speak of repetition, we normally imply a sequence in temporal order; what we do now, we may do after a lapse of time, no matter how long. This is replication of

Samples and Sampling

In Chapter 15 we dealt with the statistical properties of an arbi­trarily selected group of numbers, which can be considered a set. Every element of the set, in most cases, contributes toward deter­mining the required property. If the population of the set is too large to handle, or if economy in terms of time

Notions of Set

Set: is a collection of elements Capital letters (A, B, C . . .) are customarily used to symbolize sets. Elements: may be numbers, measurements, materials of any kind, plants, animals, men, or any other objects or entities. Lowercase letters (a, b, c, . . .) are customarily used to symbolize elements. X =

Permutations and Combinations

After doing sampling, the result of which is the set of a smaller number of elements, the relevant questions are In how many different ways can we arrange the given (small) number of elements of the set, deal­ing with all of them together? How many different subsets can we form out of a given

Quantitative Statement of Randomization

To illustrate this point, we deliberately work with a small set. Consider a population of six elements, a, b, c, d, e, and f out of which we want to have a random sample of two elements. The possibilities are fifteen subsets: This is the same as the number of combinations among six things,

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Sampling Methods

The following are typical, but by no means the only, methods used for sampling. Though there is wide variation among the methods used to suit the circumstances, there are some principles to which all sampling methods should conform to render the sam­ple worthy of statistical analysis. The sample should be representa­tive of the population,

Inference from Samples

The very idea of sampling is meaningful only when the popula­tion is large. When the population is very large, sampling is done to economize on time and expense. When the population involved is not only very large but also not practically accessible, as, for example, when finding the average height of tenth-grade boys in

Theoretical Sampling Distribution of X

When we encounter finite but large or infinite populations, sam­pling is inevitable. Suppose we are interested in knowing the pop­ulation mean, the only connection we have with the population being the sample; we have to depend on the sample mean, which is not an exact but a “probabilistic” value, meaning it is reason­ably close

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Central Limit Theorem

Now that we have looked at an example of theoretical sampling distribution in action, we are now ready to witness one of the most far-reaching generalizations in the area of statistics. Named the Central Limit Theorem, it may be stated as follows: For a given population, finite, large, or infinite, whatever may be the

Standard Normal Distribution

The normal distribution curve, as mentioned in Chapter 15, occupies a preeminent position in statistics. That the frequency distributions of many (though not necessarily all) natural and man-made random variables approximate to this shape is an important reason for its significance, which is further enhanced, as we have seen, by the consequence of the

Frequency Distribution and Probability Function

Let us say that statistics were required on the height of adult males in the United States. The usual, established, statistical pro­cedures take over. Because, obviously, it is not practical to reach every adult male in the country, an appropriate random sampling is adapted. The hypothetical data obtained and tabulated in Table 17.4 is

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Standard Normal Curve

The standard normal curve is, in fact, a generalization of many— as many as we may encounter—normal curves showing probabil­ity density distributions. Before we move to the generalization involved, it is relevant to examine the nature of the normal distri­bution curve. Firstly, it is not a frequency polygon, even if hun­dreds of straight lines

Questions/Answers Using the APSND Table

Now that we are acquainted with the numbers in Table 17.5, we may try to answer the questions we posed before and related ones. A standard normal curve implies normal distribution. This means the data we have on hand should conform, in terms of frequency distribution, reasonably closely to normal distribution. If the population

Guiding Principles

Before building, whether a bridge or a machine, planning is nec­essary. Likewise, before conducting it, an experiment needs to be planned. A well-planned experiment should be the shortest means (both in terms of time and money) to the ends, answering specific questions asked in the form of hypotheses. Planning for an experiment with quantitative

Some Preliminaries for Planned Experiments

The discussion in this chapter is limited to the so-called simple comparative experiments. The purpose of such experiments is to bring about “improvements” in the dependent variable (mea­sured as the experimental response) by causing the planned changes in the independent variables. A comparison is made between the two means (of the numerical values), one

Null and Alternate Hypotheses

Asking the questions and forecasting the likely answers from the experiment are done in the formal language of statistics. The question-answer formats are known as null and alternate hypoth­eses. These are described below. 1. Null Hypothesis in an Experiment The term null connotes negation, that is, rejecting (or saying no to) a specific assertion.

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