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

MCom I Semester Statistical Analysis Test Significance Small Samples Study Material Notes: Need of Separate Analysis for Test of Significance in Case of Small Samples Significance Testing in Small Samples Assumptions for Test Properties Of Distribution Table Significance of the Difference Between Independent sample means  Test in Paired Samples The Difference Test Method  :

CTET Paper Level 2 Previous Year Science Model paper II in Hindi

Test of Significance: Small Samples

It is very difficult to make a clear-cut distinction between small samples and large samples. However, from a practical point of view, in most of the situations, a sample is termed as small if n < 30.

Need of Separate Analysis for Test of Significance in Case of Small Samples

So far, we have discussed problems related to large samples. The large sample theory are based on the following two assumptions viz.,

(i) Sample standard deviation is close to population standard deviation and as such can be used in its place for the computation of standard error. Thus, in the computation of the standard error of the mean, the standard deviation of the sample is used in the absence of the standard deviation of the population.

(ii) The distribution of sample statistics is normal. Because of this it is possible to assign limits within which the difference between sample statistics and population parameters is likely to lie. These assumptions do not hold good when the size of the sample is small. Series of means or standard deviations of the small samples may or may not be normally distributed and similarly it is not possible to substitute the mean or the standard deviation of a small sample in place of the parameter mean or standard deviation for the calculation of standard errors. In fact, for small values of n (number of items included in the sample) the standard deviation of the sample is subject to a definite bias, tending to make it consistently lower than the standard deviation of the population. Thus, if the standard deviation of a small sample is used in the computation of the standard error of the mean, the result will also have a downward bias. It can, therefore, be said that the methods, discussed so far, when applied with small samples, the sampling errors to which our estimates are subject, are consistently under-estimated. This under-estimation of the sampling error takes away a part of its utility for purposes of statistical inference. Under such circumstances, the analysis of small samples has to be done by techniques which are different from those applicable in case of large samples. It should be clearly understood that the methods and the theory of small samples are applicable to large samples, but large sample theory cannot be used for small samples.

While dealing with small samples our main interest is not to estimate the population values as is true in large samples; rather our main object is to test a given hypothesis. In other words, we try to ascertain whether ambling fluctuations. Moreover, we should use relatively wide confidence intervals as the results of small samples use as the results of small samples usually vary widely from sample to sample. For example, if a sample of ten units gives us the mean height of 64 we cannot estimate from this the mean height of the universe. However, we can find out if this height of 64″ is consistent with the hypothesis that the true height of the universe is 66″.

In various economic and social studies usually the size of the samples is fairly large so that the techniques discussed in previous chapter are at once applicable to, them. But in such cases, where the data are collected by very costly laboratory or field experiments which require a large expenditure and considerable time it is not worthwhile and in many cases not possible, to obtain large sized samples. In all such cases it is necessary to use some dependable technique of the analysis of small samples.

Small Samples and Statistical Measures

Before dealing with the technique of actual analysis of small samples we shall briefly discuss some of the effects of the smallness of the sample size on various standard measures. These effects are as follows:

(1) The smaller the sample the greater would be the variation in the value of means of different samples. Thus in cases of small samples though the sample means may vary on either side of the true mean symmetrically yet the range of their variation would be greater than the range in case of large samples.

(2) The standard deviation of the smaller samples tends to be smaller than true standard deviation of the universe and the smaller the sample the smaller would be the standard deviation.

(3) Since the standard deviation of small samples tends to be smaller than the standard deviation of the universe, all errors based on these smaller standard deviations also tend to be small. As such unless corrections are made for the errors of small samples they would tend to underestimate the actual error.

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Arithmetic Mean and Standard Deviation in Case of Small Samples

The two important measures for any samples are the mean and the standard deviations. The mean of a small sample is found by the usual method but in case of standard deviation some different method is used. In case of small samples the standard deviation is calculated by the formula

We have mentioned in earlier chapters, the concept of degrees of freedom and have also noted that in the calculation of standard deviation, the denominator should be n – 1 and not n; but in case of large samples the use of n in place of does not involve any significant error and as such the standard deviation in of large samples is calculated by and the standard error by Fr. In case Il samples, however, the use of n in place of n – I would make a significant and as such the standard deviation in case of small samples is calculated

Significance Testing in Small Samples

Various significance tests have been developed for dealing with problems of small samples. The t-test developed by Sir William Gusset (pen name Student) and z-test developed by R.A. Fisher are very important tests and deserve special mention in context of sampling analysis for small samples. These tests are used to determine the reliability of the small samples. The basic fundamental assumptions in all the small sample tests are:

(1) The parent population(s) from which the sample(s) is are) drawn is(are) normally distributed.

(11). The sample(s) is(are) random and independent of each other. If there is drastic departure from normality in the distribution of the population (U or J shaped) from which the sample is drawn, the methods discussed below cannot be applied for good results. Since in many of the problems it becomes necessary to take a small size sample, considerable attention has been paid in developing suitable tests for dealing with problems of small samples.

We now discuss the important significance tests developed in context of small samples:

1 Tests of Significance Based on t-distribution

The pioneering work in the development of the Small Sample Theory was done by an Irish brewery employee William S. Gossett in the beginning of 20th century.

Gossett was employed by the Guinness Brewery in Dublin, Ireland, which did not permit employees to publish research findings under their own name. So Gossett adopted the pen name ‘Student’ and published his findings under this name in 1908. Thereafter, the 1-distribution is commonly called Student’s t-distribution or simply Student’s distribution.

The 1-distribution is used when sample size is 30 or less and the population standard deviation is unknown.

If X1, X,… X, be any random sample of size n drawn from a normal or approximately normal population with mean (and variance (o’) then the student’s 1-distribution is defined by the test statistics as

C = a constant required to make the area under the curve equal to unity,

v=n-1, the number of degrees of freedom. Assumptions for t-test

(1) The population from which a sample is drawn is normal or approximately normal.

(2) The observations are independent and the samples are randomly drawn samples.

(3) The population standard deviation (p) is not known.

Properties (Features) of t-distribution

(1) The student’s r-distribution like the normal distribution is bell-shaped frequency curve.

(2) The 1-statistic ranges from minus infinity to plus infinity

(3) Just like the standard normal distribution, 1-distribution is also symmetrical with the difference between mean, mode and median equal to zero, except that i-distribution with one degree of freedom has no mean.

(4) The variance of 1-distribution is greater than one and as the sample size increases it tends to move towards unity.

(5) -distribution can be used even in case of large sample but the large sample theory cannot be used for small sample.

The -Table

The 7-table given at the end is the probability integral of -distribution. It gives, over a range of values of v, the probabilities of exceeding by chance value of t at different levels of significance. The 1-distribution has a different value for each degree of freedom and when degrees of freedom are infinitely large, ther-distribution is equivalent to normal distribution and the probabilities shown in the normal distribution table are applicable. The table value oft is compared with the calculated value of 1. If calculated value of is less than the table value, then X-u is not significant; in case it is more, the difference is significant.

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Applications of the t-distribution

The student’s t-distribution test is used in the following fields :

(A) To test the significance of the Mean of a Random Sample

(B) To test the significance of the difference between Two Independent Sample Means

(C) To test the significance of the difference between Two Dependent Sample Means

(D) To test the significance of an Observed Correlation Coefficient

(A) To Test the Significance of the Mean of a Random Sample

In order to determine whether the mean of a random sample drawn from a normal population differ significantly from the hypothetical value of the population mean: the following procedure is adopted to apply the t-distribution test:

(1) Null Hypothesis (H) : There is no significant difference between the sample mean and the population mean. In other words, the difference between X and u is due to fluctuations of sampling.

Two samples are said to be dependent when the elements to those in the other in any significant or meaningful manner. dent, when the elements in one sample are related, may consist of pairs of observations made on the same objects, in cant or meaningful manner. Infect, the two samples generally, on the same selected population elements. When selected population elements. When samples are dependent they comprise the same number of elementary units. Let us now take a particular situation where :

(1) The sample sizes are equal, i.e., n = n2 = n (say), and

(ii) The sample observations (x1,x2,…,x) and (1• Y… Yn) are completely independent but they are dependent in pairs, i.e., the pairs derivations (1.Yı), (x, y2), …. (x, y) correspond to the 1st, 2nd, …, nth unit respectively. Suppose we want to find if the advertisement is really effective in promoting sales of a particular product. Let X, X,…, 2., be the sales of the product in n departmental stores for a certain period before advertisement campaign and let yı-Y2….. Yn be the corresponding sales of the same product and for the same period in the same departmental stores respectively after the campaign. Now (x,y:), i = 1,2,…, IS the pair of sales in the ith departmental store before and after the advertisement. In order to test the significance of the difference between the sales, x and y, we can’t apply the difference of means test since the samples are not independent. In this case, we apply paired t-test.

We may carry out some experiments, say, to find out the effect of training on some employees, find out the efficiency of a coaching class or determine whether there is a significant difference in the efficiency of two drugs-one made within the country and another imported.

In short, if the significance of the difference between results of two activities on the same items is to be tested, the then paired t-test is applied.

The following procedure is adopted to apply paired 1-test:

(i) First of all we find the difference (D) between the paired values of two activities. For example, V1 – x1, ()2 – X»), V3 – 13)…Vn – xn), i.e., we subtract the first value from the corresponding second value. If the second value exceeds the corresponding first value then the obtained difference will be positive and negative in the reverse situation.

(ii) Compute the mean of the differences obtained in step (i), i.e.,

D=20The value of D may be a whole number or in a fraction.

If the value of D is a whole number then we proceed with the procedure as follows:

(iii) Find out the deviations from D, i.e., d = D-D.

(iv) Square these deviations and obtain the total, i.e., d.

(v) Find out the standard deviation of the difference (S) by applying the following formula:

(D-D) Sen-1 or n-1 If the value of D is infraction then the following procedure is agented

(iii) Each item of the difference is squared up (D4) and their total EDP is obtained

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