MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes ( Part 2 )

MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes ( Part 2 ) : Standard Error of the Other Statistics Difference Between Coefficient of Correlation of Universe and Large Sample Standard Error of Coefficient Correlation Test of Significance in Attributes Assumption of tests of Significance in Attributes  :

MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes

 (2) Standard Error of the Standard Deviation

As stated earlier, the standard error of the sampling distribution of standard deviation is equal to its standard deviation. In other words, standard error of the standard deviation gives the range of deviation from the population standard deviation within which standard deviation of infinite number of large samples would lie. With the help of S.E. of Standard Deviation, we can find out whether there is a significant difference between the standard deviation of sample (os) and standard deviation of population (Op). The formula for calculating standard error of standard deviation is as follows:

Illustration 19.

(i) Find out the standard error of the coefficient of variation when mean of distribution is 5, Standard Deviation 2.345 and size of sample is 50.

(ii) Find out the standard error of the Coefficient of Scenes when size of sample is 100. Solution (1) Coefficient of Variation = x 100 = 2.345 x 100 = 46.9%

Test of Significance in Attributes

As distinguised from variable where quantitative measurement of a phenomenon is possible, in case of attributes we can only find out the presence or absence of a particular characteristic. In other words, in the sampling of attributes the given population is divided into two mutually exclusive classes, one possessing a particular attribute under study and the other not possessing that attribute. 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 attribute smoking, a sample may be drawn and its members are classified as smokers and non-smokers. On the basis of significance test of such a sample we can find out whether there is significant difference between the sample proportion and universe proportion.

In case of sampling of attributes we have to consider the following three types of problems :

(i) The ‘parameter’ value is given and it is only to be tested if an observed ‘statistic’ is its estimate;

(ii) The ‘parameter’ value is not known and we have to estimate it from the sample;

(iii) Examination of the reliability of the estimate i.e., the problem of finding out how far the estimate is expected to deviate from the true value for the population.

Use of Notations in Sampling of Attributes

Selection of an individual in sampling may be called an ‘event’ or a ‘trial’ and the total number of events are represented by ‘n’. We shall call the possession of the attribute by an individual selected as success and the non-possession as ‘failure’. Proportions of ‘success’ and ‘failure’ are represented by p and q respectively. Total number of units related to universe is represented by N. For example, if we have taken a sample of 1,000 persons from an universe of 10,000 persons and out of the selected persons we find that 400 smokers and 600 are non-smokers. These facts can be shown as follows:

Assumptions of Tests of Significance in Attributes

Before proceeding further we shall lay down certain assumptions, which we presume, would hold good in the sample which is under study. The sampling which satisfies these assumptions would be called ‘Simple Sampling’. Thus by simple sampling we shall mean a random sample in which the following conditions hold good :

(1) The probabilities of drawing individuals with attributes A, or the chance of success of various events are independent whether previous trials have been made or not. It means that the proportion of A’s at each draw of a sample unit is identical. This assumption holds good in case of tossing a coin or drawing a ball of a card provided that before the second and subsequent draws the ball or card drawn previously is replaced. Thus the probability of a coin falling ‘heads’ is identical in all throws and similarly the probability of drawing a red ball from a bag containing four red and four white balls is identical for all draws provided replacement each time.

Large Samples Study Material

(2) The probability (or p) of drawing an individual with attribute A remains constant and is the same for all samples. This condition would hold good only if the proportion of A’s in the universe remains constant each time a sample is drawn. If a dice is tossed at two different places or at two different times the probability of success (if coming of no. 6 is taken as success) would be identical. This cannot be said about sampling of attributes if the two samples have been drawn at two different places of the same universe) or at two different times. The proportion of blind in the same universe would not be identical either at two places or at the same place at different times. In the analysis of sampling of attributes we presume that this would be so.

X and o in Simple Sampling of Attributes

The binomial distribution is concerned with two numbers or probabilities. Sampling theory using proportions applies this where the population has binomial properties, in other words where the population can be divided into two distinct parts, one possessing a certain property and other not possessing that property.

We have already mentioned in earlier chapters that if the probability of the happening of an event in one trial is known, we can find the probability of its happening r times in n trials by the expansion of a binomial. Hence, the sampling distribution of the number of success, being a binomial probability model, would have its mean (u) = np. the variance (o) = npq and the standard deviation (©) = Vnpq.

Various Tests of Significance for Attributes

The various types of significance test may be studied under the following heads :

(A) Test for Number of Successes,

(B) Test for Proportion of Successes, and

(C) Test for Difference between Proportions

21.28

Here, it would be appropriate to explain the meaning of standard error related to the simple sampling of attributes.

Simple Sampling of Attributes and Standard Error

The standard deviation of simple sampling is briefly called Standard Error. The term standard error has in reality a wider meaning than merely the standard deviation of simple sampling. But for the sake of convenience, the term can be defined as mentioned above.

(A) Test for Number of Successes

In such a case we have to find out whether there is a significant difference between the expected number of successes and observed number of successes. For this purpose, we shall calculate the standard error of 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 = Vnpg where n = size of sample

p = Probability of success in each trial 9 = (1-P), i.e., probability of failure Interpretation may be based on any one of the following criteria :

(1) Significance Ratio Criteria : If the difference between the actual and observed frequencies is more than three times the standard error, the difference is said to be significant which means that such a difference could not have arisen due to fluctuations of sampling or the probability of such a difference arising due to chance is very very low. If the difference is less than three times the standard error it could have arisen due to fluctuations of sampling. We can also test the results at 5% level of significance and at 1% level of significance by 1.96 standard error and 2.5758 standard error respectively.

(2) Fiducially Limits Criteria : For significance testing purpose we generally make use of the criteria np + 3vnpq. If the observed value of the number of successes is within np + 3vnpq limits of the expected number of successes, the difference between the observed value and the expected value is considered insignificant and is presumed to have arisen due to sampling fluctuations but if the observed number of successes does not fall within the said limits, the difference is taken as significant and could not have arisen due to sampling fluctuations. The significance of the difference between observed number and expected number of successes at 5 percent level of significance and at 1 percent level of significance can as well be judged by the criteria 1.96Vnpq and 2.5758Vnpq respectively. The following examples make all this quite clear.

Illustration 20.

(i) A coin is tossed 10,000 times and head turned to be 5,195 times. Test the hypothesis that the coin is unbiased.

(ii) In 400 tosses of a coin 220 heads and 180 tails appeared. Determine the limits for the expected frequency of head.

(iii) A sample of 600 persons was selected at random from Bangalore city and 53% are found to be male. Is there any reason to doubt the hypothesis that male and female are equally distributed ?

Large Samples Study Material