Here we shall consider sampling from a population which is divided into two mutually exclusive and collectively exhaustive classes one class possessing a particular attribute, say A, and the other class not possessing that attribute, and then note down the number of persons in the sample of sizen, possessing that attribute. The presence of an attribute in sampled unit may be termed as success and its absence as failure. In this case a sample of n observations is identified with that of a series of n independent Bernoulli trials with constant probability P of success for each trial. Then the probability of x successes in n trials, as given by the binomial probability distribution is
If X is the number of successes in n independent trials with constant probability P of success for each trial
E(X) = nP and V(X) = n PQ
Where Q = 1- P, is the probability of failure.
It has been proved that for large n, the binomial distribution tends to normal distribution. Hence for large n, X≈ N(nP, nPQ) that is
And we apply the normal test.
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