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Tests of
Contents:
hypothesis
Tests of significance for small samples
Student’s t- test
Properties of t-Distribution
The t- table
Application of the t-Distribution
To test the significance of
Mean of a random sample
Means of two samples (independent)
Means of two samples (dependent)
Observed correlation coefficient
Z-transformation
Limitations of tests of significance
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Tests of significance for small samples
Student’s t-Distribution


The t-distribution is used when sample size is 30
or less and the population standard deviation is
unknown.
The “t-statistic” is defined as :
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Properties of t-Distribution
The variable t- distribution ranges from
minus infinity to plus infinity.
The constant c is actually a function of , so
that for particular value of , the
distribution of (t) is completely specified.
Thus (t) is a family of functions, one for
each values of .
The t-distribution is symmetrical and has a
mean zero.
The variance of it is greater than one.
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Variance approaches one as the number
of degree of freedom and the sample size
becomes large.
COMPARISON OF NORMAL & T-DISTRIBUTION
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The t-Table
The t-table is the probability integral of tdistribution.
a) Over a range of values of , the
probability of exceeding by chance value
of t at different levels of significance.
b) The t-distribution has a different value
for each degree of freedom.
c) When degrees of freedom are infinitely
large, the t-distribution is equivalent to
normal distribution & values given in the
table are applicable.
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Applications of t- distribution
1.
To test the significance of mean of
random sample
d= deviation from the assumed mean.
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Example: a random sample of size 16 has 53 as mean. The sum of squares
of the deviations taken from mean is 135. can this sample be regarded as
taken from the population having 56 as mean ?
Solution. Let us take the hypothesis that there is no significant difference
between the sample mean and hypothetical population mean. Applying t
test :
t = XS- µ √n
µ=56, n= 16 ∑(X-X) = 135
X=53
S= ∑(X-X) = 135 = 3
n-1
t=
53-56
3
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√16 =
3*4
=
3
4
= 16-1 = 15. for = 16 , t 0.05 = 2.13.
The calculated value of t  the table value.
The hypothesis is rejected.
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Testing difference between means of
two samples (independent samples)
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When actual means are in fraction
the deviation should be taken from
assumed mean
The degree of freedom = (n1-n2-2).
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If number of observations
and standard deviation of
two samples is given:
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Example : the mean life of a sample of 10 electric light bulbs was
found to be 1,456 hours with standard deviation of 423 hours.
A second sample 17 bulbs chosen from a different batch showed
a mean life of 1,280 hours with standard deviation of 398
hours. Is there a significant difference between the means of
the two batches ?
Solution : let us take the hypothesis that the means of two
batches do not differ significantly.
Applying t- test
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Testing difference between means of
two samples (dependent samples)
The t- test based on paired observation is defined by
the following formula
Note : t is based on (n-1) degrees of freedom.
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Example : a drug is given to 10 patients, and the
increments in their blood pressure were recorded to be 3,
6, -2, 4, -3, 4, 6, 0, 0, 2. is it reasonable to believe that the
drug has no side effect on change of blood pressure ?
Solution : let us take the hypothesis that the drug has no
effect on change of blood pressure. Applying the
difference test:
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The calculated value  the table value.
The hypothesis is accepted .
Hence it is reasonable to believe that drug has no effect on
change of blood pressure.
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Testing the significance of an
observed correlation coefficient
If we are to test the hypothesis that the correlation
coefficient of population is zero, we have to apply
the following test :
Note : Here t is based on (n-2) degrees of freedom
If calculated value of t exceeds t0.005 for (n-2) , we say that
the value of r is significant at 5% level.
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Example : A random sample of 27 pairs of observations from
a normal population gives a correlation coefficient of 0.42.
is it likely that the variables in the population are
uncorrelated ?
Solution : let us take the hypothesis that there is no
significant difference in the sample correlation and
correlation in population. Applying test:
For =25, t0.005= 1.708.
The calculated value > the table value.
The hypothesis is rejected.
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Z-test
of
the
significance
correlation coefficient
of
The Z- test given by Prof. Fisher is used to test:
a) Whether an observed value of r differs
significantly from some hypothetical value, or
b) Whether two sample value of r differ
significantly.
The Z-test is defined as following:
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Limitations of tests of
significance
 They
should not be used mechanically.
 Conclusions are to be given in terms of
probabilities and not certainties.
 They do not tell us “why” the difference
exists.
 Serious violation of of assumptions.
 Non-publication
of
non-significant
results.
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Thank you
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