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 before and one after the experiment. Whether the intended improvement did or did not take place is decided based on such a comparison. Also, such a comparison is often used for selecting one of the available two populations with a certain parameter as the criterion, common to both the populations, measured and expressed in numbers.

**1. Sample Size**

When the populations are very large or unlimited, sampling is necessary. On the other hand, when the population is limited or reasonably small, sampling becomes less necessary. In planned experiments, every replication of the experiment with the altered (or adjusted) independent variables provides one value of the (hopefully) improved dependent variable. The collection of a number of such values constitutes the “size” of the sample. The larger the sample size demanded, the more replications required. Too many replications result in unnecessary expense, and too few are inadequate for getting a dependable value of the sample mean. The right sample size, thus, needs to be found from statistical considerations.

However, exceptional situations exist. For example, in a hypothetical case, the experimenter, altering the independent variable serving as the “cause,” can bring about an increase in the pressure, the “effect,” in a steam chamber. Once the independent variable is adjusted as planned, pressure in the steam chamber simply needs to be recorded over an extended period of time (to neutralize the effect of uncontrolled variables, also known as noise.) Then, with one replication, a large number of pressure readings can be obtained; these serve as the population. Randomized sampling from such a population needs to be done strictly following statistical considerations. But this does not involve the expense of a large number of replications.

Computation of the sample size, either as the number of replications or as the number of elements randomly selected from a population, involves the following factors:

- α and β risks
- δ, the increment of improving a given property as the dependent variable
- σ, the standard deviation, either known or unknown
- The nature (or statement) of the null hypothesis for the experiment

Formulas for computing the sample size to suit different experimental situations vary. The experimenter needs to use the formula prescribed by statisticians for a specific set of experimental situations.

**2. Minimum Acceptable Improvement**

In a planned experiment, the degree of improvement measured as the enhanced value of the dependent variable needs to be stated.

For instance, in a planned experiment for upgrading the life of a given ball bearing in a specific application, the goal should not be “to improve the life of the bearing to the maximum extent possible.” Instead, it is desirable, for example, to state, “to improve the life from the present average of ten thousand hours to the desired average of eleven thousand hours of satisfactory life.” All these ideas are implied in what are known as the hypotheses, which are stated in symbolic form, containing none of the above words.

Source: Srinagesh K (2005), *The Principles of Experimental Research*, Butterworth-Heinemann; 1st edition.

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