Tests of Significance for Attributes

As distinguished from variable where quantitative measurement of a phenomenon is possible, in case of attributes we can only find out the presence or absence of particulars characteristics. The sampling of attributes may, therefore, be regarded as the drawing of samples from a population whose members possess the attribute A or not A. For example, in the study of attributes ‘Literacy’ a sample may be taken and people classified as literates and illiterates. With such data the binomial type of problem may be formed. The selection of an individual on sampling may be called ‘event’, the appearance of an attribute A may be taken as ‘successes and its non-appearance as ‘failure’. Thus if out of 1,000 people selected for the sample, 100 are found literates, and 900 illiterates, we would say that the sample consists of 100 units out of which 100 are successes and 900 ‘failure’. The probability of success or P = 100/1,000 or 0.1 and the probability of failure or q = 900/1,000 = 0.9 so that p + q = 0.1 + 0.9 = 1.

The various tests of significance for attributes are discussed under the following heads:

(i)    Test for Number of Successes,

(ii)    Test for proportion of Successes, and

(iii)    Test for Difference between Proportions.

(i)    Test for Number of Successes

The sampling distribution of the number of successes follows a binomial probability distribution. Hence its standard error is given by the formula:

S.E. of number of successes = √nqp

            Where n = size of sample

                p = probability of success in each trial

                q = (1-p), i.e., probability of failure.

(ii)    Test for Proportion of Successes

Instead of recording the number of success in each sample, we might record the proportion of successes, that is 1/nth of the number of successes in each sample. As this would amount to dividing all figures of the record by n, the mean proportion of successes by p, and the standard deviation of the proportion successes √pq/n. Thus we have the following formula:

    S.E.p = √pq/n.

(iii)    Test for Difference between Proportion

If two samples are drawn from different population, we may be interested in finding out whether the difference between the proportion of successes is significant or not. In such a case we take the hypothesis that the difference between p₁, i.e., the proportion of success in one sample, and p₂, i.e., the proportion of successes in another sample, is due to fluctuation of random sampling. The standard error of the difference between proportions is calculated by applying the following formula:

S.E. (p₁ – p₂) = √pq(1/n₁ + 1/n₂)

Where p = the pooled estimate for the actual proportion in the population.

The value of p is obtained as follows:

    P = n₁p₁ + n₂p₂ or p = x₁ + x₂
            _n₁ + n₂        ____n₁+n₂

Where x₁ and x₂ stand for the number of occurrence in the two samples of sizes n₁ and n₂ respectively.

If p₁ – p₂ is less than 1.96 S.E. (5% level of significance), the difference is regarded
     S.E.
as due to random sampling variation, i.e., as not significant.

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