MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes ( Part 3 )

MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes ( Part 3 ): Miscellaneous Illustration Sampling of Variables Examinations Questions Long Answer Questions Short Answer Questions Objective Questions ( This Post Most Important For MCom I Semester Students )

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(D) Significance Test of Difference between Different Proportions of Two Samples

Sometimes we may come across cases where the proportion of A’s are not the same in the two materials or universes from which the samples have been chosen, but P, and p2 are the real proportions. In such cases we may be interested in finding out whether the difference would vanish if further samples were taken. Such a situation usually arises in questions where association between attributes is studied. The proportion of A’s in the universe of B and in the universe of B may be different from each other and we can presume that Pı and p2 are the real proportions and then we can test our hypothesis. We may then find out whether further samples would also indicate the difference in the proportion of A’s in the universe of B’s and p’s or whether the difference has arisen only in the present case due to sampling fluctuations. The following examples make all this quite clear :

Example 1 : In a random sample of 1,000 literates, 600 are found to be cmployed and out of 800 illiterates, 440 are found to be employed. Here, first sample reveals the proportions of employed persons among literates whereas second sample reveals the proportions of employed persons among illiterates.

Example 2 : Out of 1.000 vegetarian persons, 200 suffer from overweight and out of 4,000 non-vegetarians 1,000 are suffering from overweight. Here, first sample reveals the proportions of overweighted persons among vegetarians whereas second sample reveals the proportions of overweighted persons among non-vegetarians.

In short, if the proportions of A’s are not the same in the two universes from which samples have been taken, then the standard error of the two proportions Pı and P2 would be respectively as follows:

If the actual difference between p, and p2 is more than three times the standard error of the difference, it is significant otherwise it may have arisen due to sampling fluctuations and further samples may not indicate any significant difference in the values of P, and P2. Illustration 27.

(i) Out of 1,000 vegetarian persons, 200 suffer from overweight. Out of 4,000 non-vegetarians, 1,000 are suffering from overweight. Is there an association between overweight and non-vegetarianism ?

(ii) In a random sample of 500 literates, 300 are found to be employed and out of 400 illiterates, 220 are found to be employed. Is the difference of employed significant ?

(iii) In a random sample of 1,000 blind persons, 400 persons are found to be literate. In another sample-sized sample of non-blind persons, the number of literate persons are found to be 600. Is this difference significant ? Solution

(i) Let us have a null hypothesis that there is no association between overweight and non-vegetarianism.

Since the difference is more than 3 times of S.E., therefore the hypothesis is rejected and there is significant difference i.e., there is an association between non-vegetarianism and over weight.

EXAMINATION QUESTIONS

Long Answer Theoretical Questions

1 What do you understand by ‘testing of significance’ ? Explain the methods used for testing the difference between arithmetic means of two large samples.

2. The standard error can be used to gauge the precision of a statistical estimate or to permit a judgment being made of the divergence between expected and observed values. Discuss clearly the concept of standard error of an estimate and its various uses in practice.

3. Why should there be different test functions for testing the significance of the difference in means when samples are (a) small, and (b) large ?

4. State the various steps in tests of significance.

5. What are the various tests of significance generally used in sampling of attributes ? Explain.

6. Discuss the main principles of large sample theory with special reference to sampling of attributes.

7. Define the standard error of a ‘statistic’. Explain the basis of large sample tests of significance based on a standard error.

8. Explain the concept and role of standard error in large sample theory.

9. Explain the terms: (a) null hypothesis, (b) level of significance. 10. “The ultimate objective of sampling is to generalize about the total population.’ Explain this statement.

10. How would you test the significance of the difference between the means of two large samples ? Explain in the context of various types of problems stating the appropriate formulae used.

Short Answer Theoretical Questions

1 What is meant by Standard Error ?

2. What do you understand by a test of significance ?

3. What is sampling distribution ?

4. Distinguish between Finite and Infinite Universe.

5. Distinguish between ‘Parameter’ and ‘Statistic’.

6. What is sampling and what are its objects ?

7. Explain the concept of ‘Null Hypothesis’.

8. Explain the concept of ‘Level of Significance’.

9. Explain the concept of ‘Critical Value.’

Objective Questions are ‘True’ or ‘False’ :

1 The value obtained from the study of samples are known as ‘statistic

(True)

2. The statistical measurement of the characteristics of all the units of universe are called parameters.

(True)

3. Tune-error is committed when the hypothesis is true but our test rejects

(True)

4. For large samples, the standard error of mean is o/vn.

(True)

5. He represent the null hypothesis.

(True)

6. When the hypothesis is true and our test accepts it, this is called Type-I error.

(False)

Fill in the blanks :

1 Type-I error is committed when the hypothesis is true but our test ………. it.

2. Type– II errors are made when we accept a null hypothesis which is ……..

3. Standard error provides an idea about the ………. of samples.

4. The standard deviation of a sampling distribution is called ……….

5. The distribution formed of all possible values of a statistic is called the ……..

Ans. (1) Rejects, (2) Not True, (3) Unreliability, (4) Standard error, (5) Sampling distribution, Select the Correct Option :