Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Testing of Hypotheses I 197 HYPOTHESIS TESTING OF MEANS Mean of the population can be tested presuming different situations such as the population may be normal or other than normal, it may be finite or infinite, sample size may be large or small, variance of the population may be known or unknown and the alternative hypothesis may be two-sided or onesided. Our testing technique will differ in different situations. We may consider some of the important situations. 1. Population normal, population infinite, sample size may be large or small but variance of the population is known, H a may be one-sided or two-sided: In such a situation z-test is used for testing hypothesis of mean and the test statistic z is worked our as under: X − μ H z = σ n p 2. Population normal, population finite, sample size may be large or small but variance of the population is known, H a may be one-sided or two-sided: In such a situation z-test is used and the test statistic z is worked out as under (using finite population multiplier): X − μH0 z = eσ p nj × bN − ngbN − 1g 3. Population normal, population infinite, sample size small and variance of the population unknown, H a may be one-sided or two-sided: In such a situation t-test is used and the test statistic t is worked out as under: and σ s X −μH t = σ / n = ∑ − s 0 2 dXiXi n − 1 b g 0 with d.f. = (n – 1) 4. Population normal, population finite, sample size small and variance of the population unknown, and H a may be one-sided or two-sided: In such a situation t-test is used and the test statistic ‘t’ is worked out as under (using finite population multiplier): X − μH0 t = with d.f. = (n – 1) eσs/ nj × bN − ng/ bN − 1g
Table 9.3: Names of Some Parametric Tests along with Test Situations and Test Statistics used in Context of Hypothesis Testing Unknown Test situation (Population Name of the test and the test statistic to be used parameter characteristics and other conditions. Random One sample Two samples sampling is assumed in all situations along with Independent Related infinite population 1 2 3 4 5 Mean ( μ) Population(s) normal or z-test and the z-test for difference in means and the test Sample size large (i.e., n > 30) or population test statistic statistic variance(s) known z X = − μ H0 σ p n z = X1 − X2 2 F 1 1 I σ p + HG n1 n2 KJ In case σ p is not is used when two samples are drawn from the known, we use same population. In case σ p is not known, we use σ s in its place calculating σ s12 in its place calculating σ s = Σd i 2 Xi − X n − 1 σ s12 = where D = X − X 1 s1 s 2 1 2 2 2 2 2 2 e j e j n σ + D + n σ + D d i d i 1 1 12 D = X − X X 2 2 12 12 n X + n X = n + n 1 1 2 2 1 2 n + n 1 2 Contd. 198 Research Methodology
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Testing of Hypotheses I 197<br />
HYPOTHESIS TESTING OF MEANS<br />
Mean of the population can be tested presuming different situations such as the population may be<br />
normal or other than normal, it may be finite or infinite, sample size may be large or small, variance<br />
of the population may be known or unknown and the alternative hypothesis may be two-sided or onesided.<br />
Our testing technique will differ in different situations. We may consider some of the important<br />
situations.<br />
1. Population normal, population infinite, sample size may be large or small but variance<br />
of the population is known, H a may be one-sided or two-sided:<br />
In such a situation z-test is used for testing hypothesis of mean and the test statistic z is<br />
worked our as under:<br />
X − μ H<br />
z =<br />
σ n<br />
p<br />
2. Population normal, population finite, sample size may be large or small but variance<br />
of the population is known, H a may be one-sided or two-sided:<br />
In such a situation z-test is used and the test statistic z is worked out as under (using<br />
finite population multiplier):<br />
X − μH0<br />
z =<br />
eσ p nj × bN − ngbN − 1g<br />
3. Population normal, population infinite, sample size small and variance of the<br />
population unknown, H a may be one-sided or two-sided:<br />
In such a situation t-test is used and the test statistic t is worked out as under:<br />
and σ s<br />
X −μH<br />
t =<br />
σ / n<br />
= ∑ −<br />
s<br />
0<br />
2<br />
dXiXi n − 1<br />
b g<br />
0<br />
with d.f. = (n – 1)<br />
4. Population normal, population finite, sample size small and variance of the population<br />
unknown, and H a may be one-sided or two-sided:<br />
In such a situation t-test is used and the test statistic ‘t’ is worked out as under (using<br />
finite population multiplier):<br />
X − μH0<br />
t =<br />
with d.f. = (n – 1)<br />
eσs/ nj × bN − ng/ bN − 1g
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