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SMALL SAMPLE TESTS
In the previous chapter we discussed tests based on large samples 11
is quite actual that the results derived hom large samples are very reliable
The value of a statistic obtained from a large sample is generally close to
the parameter of the population. But there are situations when one has to
take a small sample. if a new medicine is to be introduced a doctor
cannot test the new medicine by giving it to many patients Instead of that
He will try the medicine on a few patients. Thushe takes a small sample
Generally a sample having number of observations less than or equal to 30
is regarded as a small sample in large sample it is assumed that the
distribution of the sample statistic is approximately normal The S. Dola
A large sample can be taken as an estimate of the population S. D. These
assumptions are not true for a small sample the distribution of a statistic
obtained from a small sample cannot be considered as normal. Also the
SD) of the sample cannot be regarded as an estimate of the population
SD. So for testing the significance of the difference between sample
mean and population mean in case of a small sample the sampling
distribution of the relevant statistic should be known Fisher's 2 cte
are such distributions useful in small sample tests
The difference between large sample test and small sample testing given below.
13.2 Difference Between Large Sample Tent and Small Sample Text
Large sample tests
(1) The sample size is greater
(1) The sample size is 30 or less than 30
(2) The value of a statistic obtain
(2) The value of a statistic from the sample can be taken obtained from the sample an estimate of the cannot be taken as an population parameter estimate of the population
(3) Normal distribution is used for
(3) Sampling distribution like /
Totes are used for testing
Assumptions of ¢ distribution : me
(i) The population from which the sample is drawn is normal
(ii) The sample is random.
ii) The population $.D. is not known
Properties of (distribution :
(i) The probability curve of ¢ distribution is symmetrical
(ii) The tails of the curve are asymptotic to the x-axis.
(i) When » > 2, distribution tends to normal distribution,
(iv) The form of the s-distribution varies with the degrees of freedom
Uses of distribution = i
(i) For testing the significance of the difference between sample
mean and population mean.
(ii) For testing the difference between means of two samples.
(iii) For testing significance of the observed correlation coefficient.
(iv) For testing the significance of observed regression coefficient.
We shall now study the above mentioned uses of / distribution.
13.4 Test of Significance of a Mean of a Small Sample
Suppose a random sample x1, X2. X3... &, is drawn from a normal
population and the mean and variance of the sample are ¥ and S*
respectively. If we want to test the hypothesis that there is no significant
difference between sample mean ¥ and the assumed mean jt of the
population. We can apply for the t test in the following way.
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