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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Accepting (or Rejecting) Hypotheses: Objective Criteria

At this stage, the experiment has yet to be conducted and the experimenter has yet to decide which of the two hypotheses he should favor. But the “favoring” is not a matter of fancy; it can­not be decided arbitrarily. The consequences of accepting one or the other of the possible hypotheses should be logically

Procedures for Planning the Experiments

With minor variations, the following are the steps for designing simple comparative experiments: Identify whether O is known for the responses rep­resented in μ0. If it is not known, more calcula­tions will be required, as we will see later in the procedure. State the null and the alternate hypotheses to represent the experimental conditions.

Other Situation Sets of Experimental Research

Thus far we have dealt in detail with only, perhaps, the two most used situation sets. We have seen the difference of requiring the additional steps in the procedure for situations in which a is not known. This difference applies to all other situation sets as well. We will now visit a few more

Operating Characteristic Curve

A given set of experimental situations can be represented in a graphical form, referred to as the operating characteristic curve for that set. To show how it is derived, we will revisit the design for experimental Situation Set 1. H0: μ1 = μ0 (average pressure, 250 psi) (δ = 10; enhanced pressure, 260 psi)

Sequential Experimenting

As the reader should have noticed, the two key steps in designing the experiment in all the variations so far presented are (1) find­ing the number of items in the sample N, and (2) computing the criterion value, X, to compare it with the mean of the response output, X1. If the sample items

The Way to Inference from Experimental Data

During the life span of an experiment, following the stages of conducting the experiment and collecting data, there comes a time when the experimenter, with his logbooks loaded with mea­surements, meter readings, and various counts of other kinds, will have to sit and ponder how reliable each bunch of observa­tions (measurements, readings, or counts)

Estimation (From Sample Mean to Population Mean) of Experimental Data

As pointed out in Chapter 18, the mean is the most significant statistic of a given set. The mean of the sample set A—56.4, 57.2, 55.9, 56.2, 55.6, 57.1, and 56.6—is 56.43. This is called the sample mean, X, and when it is used as an estimator of the popula­tion mean, fl, it is

Testing of Hypothesis in Experimental Research

Testing of statistical hypothesis has several applications, almost wherever statistics is applicable. In this book, however, we shall confine the discussion to experimental research, considering only one obvious and hypothetical case of application; this will show the way to other possible applications with minor modifications as required. Let us imagine that an experiment has

Regression and Correlation of Experimental Data

When two sets of variables are associated with any kind of rela­tion, their relation can be represented on a graph as a set of points, each point determined by a pair of corresponding coordi­nates, one from each set. When there is a cause-effect relation, the values of the independent variable are shown on the

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Multiple Regression of Experimental Data

Thus far, we have dealt with simple linear regression, which is adequate for most lab experiments required in college course work. For research work, one-factor-at-a-time experiments are inadequate. It is now time for us to reflect on how close—or how far away—the reality of the experimental situation is to the results obtained by simple