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

MCom I Semester Statistical Analysis Test Significance Small Samples Study Material Notes ( Part 2 ) : Fisher Z test Variance Ratio Test F Test Calculation procedure F Test Calculation fo Variance Limitations of Test of Significance Small Samples At a Glance Small Samples Long Answer Questions Short Answer Questions Objective Questions  :

MCom I Semester Business Environment Foreign Investment Study Material notes

Fisher’s Z-test

The 7-test for correlation (r) discussed above is applicable only for determining whether the computed r is significantly different from zero. When we want to test the sample correlation against any other theoretical value of r (other than zero), or if it is desired to test whether the two given samples have come from same population or not, t-test cannot be used.

Prof. Ronald Fisher has given a method of testingthe significance of the difference between two coefficients of correlation. According to this method, the coefficient of correlation is transformed into z and hence named Z-transformation. The statistic Z given by Prof. Fisher is used to test (a) whether an observed value of r significantly differs from some hypothetical value, or (b) whether there is significant difference between the two sample values of r. For testing whetherr differs significantly from zero, the t-test is preferable.

(a) To Test the Significance between the Difference of an Observed Value of Coefficient of Correlation and Some Hypothetical Value: The following procedure is adopted to test whether an observed value of r significantly differs from some hypothetical value:

(i) First of all, the observed and hypothetical value of r is transformed into Z. The hypothetical value of r refers to the population correlation coefficient. The formula of Z-transformation is as follows:

Note: Here it should be noted that we will calculate Z separately for observed and hypothetical value of ri.e., there will be one z for observed value of r and another for hypothetical value of r. To distinguish the observed and hypothetical

As the significance ratio is less than 1.96, the difference between the two given values of coefficient of correlation at 5% level is significant and it can be concluded that the two samples come from the same population.

Variance-Ratio Test : F-Test

The various tests of significance discussed so far are not suitable for test of significance of two or more sample estimates of the universe variance. Very often we like to test whether the two independent estimates of population variance differ significantly, or whether the two samples may be regarded as drawn from the normal population having the same variance. The F-test tells us whether the two independent estimates of population variance differ significantly, or whether the two samples may be regarded as drawn from the normal populations having the same variance. This test is carried out with the help of F-ratio.

The F-test is named in honor of the great statistician R.A. Fisher. F-test initially was used to verify the hypothesis of equality between two variances. Infact, F-test is a test of significance concerning two sample variances. It is based on F-distribution and is concerned with the F-ratio (or the variance ratio) rather than the difference between variances. Prof. Snedecor’s name is also associated with this test. This test has been explained in detail in a separate chapter entitled “Analysis of Variance”.

Assumptions of F-Test :

The F-test is based on the following assumptions :

(1) Normality: The populations for each sample must be normally distributed with identical mean and variance.

(2) Independence: All sample observations must be randomly selected and independent.

(3) Homogeneity: The variance within each group should be equal for all groups (cí = oś= … = This assumption is needed in order to combine or pool the variances within the groups into a single ‘within groups’ source of variation.

(4) Independence of Error : It states that the error (variation of each value around its own group mean) should be independent for each value.

(5) Above One : Variance ratio should always be equal to or greater than 1. This is the reason that larger variance is divided by the smaller variance.

(6) Never be a Negative Value : Since the F-distribution is always formed by a ratio of squared values, it can never be a negative value.

Conclusion: Since computed value of F = 1.038 is less than the table value of F = 3.39, so difference is not significant at 5% level of significance. Hence, null hypothesis is accepted and it may be concluded that two samples have been drawn from the same normal population.

Limitations of Tests of Significance

A number of tests are carried out for determining the significance of statistic in large and small samples. On the basis of test results many important decisions are made. But there are several limitations of these significance tests which should always be borne in mind. Important limitations are as follows:

(1) The Tests of Significance should not be used Mechanically : It should be kept in view that tests of significance are simply the raw materials from which to make decisions, not decisions in themselves. There may be situations where real differences exist but do not produce evidence that they are statistically significant or the other way round. Hence, proper interpretation of statistical evidence is important to intelligent decisions.

(2) The Statistical Results cannot be said to be Infallible proof of a Hypothesis: Statistical inferences based on the significance test cannot be said to be entirely correct evidences concerning the truth of the hypothesis. This is especially so in case of small samples where the probability of drawing erring inferences happens to be generally higher. For greater reliability, the size of samples be sufficiently enlarged.

(3) Conclusions are to be given in terms of Probabilities and not Certainties : The results of significance tests are based on probabilities and as such cannot be reseed with full certainty. When a test shows that a difference is statistically indicant then it simply suggests that the difference is probably not due to chance sample having smaller variance. Interpretation of F-Test : Difference between the variance of two samples will be significant, if calculated value of F> table value of F at desired level of significance and connected d.f.

EXAMINATION QUESTIONS

Long Answer Theoretical Questions

1 What is meant by Y’ test and Z test ? Discuss their utility in statistics.

2. What do you understand by ‘Z’ test and 7 test ? Describe their practical significance.

3. What do you understand by Fisher’s F-test and Students t-test ? Indicate some practical applications of these tests.

4. Explain the t-test for testing the significance of difference between two sample means. State clearly the assumptions involved.

5. What do you understand by r-test, F-test and Z-test ? Describe their practical importance.

6. Discuss the F-test for testing the equality of two sample variances. State clearly the assumptions involved in F-test. innately of

7. Explain how the Student’s 1-distribution is used to test the significance of diligence between the means of two samples stating clearly the underlying, assumptions.

8. Explain Fisher’s transformation of the correlation coefficient and indicate its use in tests of significance.

9. Explain the difference between 1-distribution and F-distribution, discussing their properties, uses, assumptions, forms and shapes.

10. Why should there be different formulae for testing the significance of the difference between means, when the samples are :

(i) small and (ii) large ?

11. What do you understand by a test of significance ? Describe the steps involved in testing the significance of an observed correlation coefficient?

Short Answer Theoretical Questions

1 ‘Discovery of Students 7 is regarded as a landmark in the history of Statistics.” Examine.

2. Write short notes on the following:

(i) Limitations of the tests of significance;

(ii) Degrees of freedom,

(iii) Fisher’s Z-Test,

(iv) F-Table,

(v) Level of significance,

(vi) Z-Transformation,

(vii) Critical value,

(viii) Null hypothesis.

3. Explain how the Student’s 1-distribution may be used :

(a) To test the significance of the sample correlation coefficient in a sample drawn from a vicariate normal population;

(b) To test the significance of the difference between the mean yields of any two varieties in an agricultural experiment.

4. Assuming the samples to be small, state which test would you apply in the following cases. Also write down the procedure of testing and the formulae to be applied :

(i) The production capacity of workers working in a section of a factory, has significantly increased or not as a result of one months’ intensive training. (ii) The difference between correlation coefficients in case of two samples is significant or not. (iii) On the basis of data relating to two samples, testing the hypothesis (a) that the variances of two parent populations are equal and (b) the difference between means of two populations is not significant.

Objective Questions are ‘True’ or ‘False’:

1 Sample, in which number of units is less than 30, is called small sample.

(True)

2. Law of Inertia of large numbers work appropriately with regard to small samples.

(False)

3. If calculated value of F > F0.05, then H, is true.

(False)

4. If calculated value of Y’ < 70.05, then H, is true.

(True)

5. 7-Test is associated with the name of ‘Karl Pearson’.

(False)

6. The samples mean () is the best estimator of the population mean

7. Student ‘-distribution is the frequency curve of bell shaped like normal distribution.

(True)

8. 7-distribution is a continuous distribution.

(True)

9. Coefficient of correlation is converted into Z-statistic under ‘Z’-Test. (True)

10. F-Test is also known as Variance-Ratio Test.

(True) Fill in the blanks :

1 Sample, in which number of units is less than ………., is called small sample.

2. F-Test is associated with the name of

3. Student 7-distribution is the frequency curve of ………. shaped like normal-distribution.

4. If calculated value of F > F0.05, then H, is ……….

5. Under Z-Test, we make the test of significance for 6. F-Test is also known as ……….. Test.

Ans. (1) 30, (2) Ronald A. Fisher, (3) Bell, (4) False, (5) Coefficient of correlation, (6) Variance ratio.