Condition of T-test used

Test of significance of a single mean.

Under H₀, the test statistic is.

$$t=\frac{Difference}{S.E\overline{(X)}}$$

$$\frac{\overline{X}-µ}{\frac{s}{\sqrt{n}}}$$
$$\frac{\overline{X}-µ}{\sqrt{\frac{s^2}{n}}}∼t_n=1$$

and it followa students t-distribution with (n-1)degree of freedom

$$\overline{X}=Sample\,mean=\frac{∑X}{n}$$

$$S^2=an\,unbiased\,estimate\,of\,the\,population\,varience\,and\,it\,is\,computed by$$

1. Actual mean method.

$$S^2=\frac{1}{n-1}∑(X-\overline{X})^2$$

$$=\frac{1}{n-1}(∑X^2-n\overline{X^2})$$

This is application when the mean value is in whole number.

2.Direct method.

$$S^2=\frac{1}{n-1}[∑X^2-n∑\overline{X^2}]$$

$$S^2=\frac{1}{n-1}[∑X^2-\frac{(∑X)^2}{n}$$

$$\frac{(∑X)^2}{n}$$

This is applicable when the mean value is in fractional form and the data are given at most two digits.

3. Shortcut or assumed mean method

$$S^2=\frac{1}{n-1}∑X^2-\frac{(∑X)^2}{n}$$

$$where\,d=X-A$$

$$and\,\overline{X}=A+\frac{∑d}{n};A=assumed\,mean$$

when the biased estimate of the population variance i,e s2 or standard deviation 's' given then the value of t is computed by.

$$\frac{\overline{X}-µ}{\sqrt{\frac{s^2}{n-1}}}∼t_n=1$$

$$\frac{\overline{X}-µ}{\sqrt{\frac{s^2}{n-1}}}∼t_n=1$$

$$where\,sample\,variance\,S^2=\frac{1}{n}∑(X-\overline{X})^2$$

$$=\frac{1}{n}[∑X^2-n\overline{X^2}]$$

$$\frac{∑X^2}{n}-\frac{(∑X)}{n}^2$$

Test of significance of a difference between two means.

Under the assumption thatσ₁²=σ²₂=σ² i,e population variances are equal but unknown the test statistics under H₀:µ₁=µ₂ is

i,e the test statistic t follows t-distribution with n1+n2-Z degree of freedom.

$$where\,\overline{X_1}=mean\,of\,first\,sample=\frac{∑X_1}{n}$$

$$where\,\overline{X_2}=mean\,of\,first\,sample=\frac{∑X_2}{n}$$

$$S_p^2=an\,unbiased\,estimate\,of\,the\,common\,population\,variance\,and\,its\,value\,is\,computed\,by\,the\,following\,methods$$

Reference

Kerlinger, F.N. Foundation of Behavioural Research. New Delhi: Surjeet Publication, 2000.

Kothari, C.R. Research Methodology. India: Vishwa Prakashan, 1990.

Singh, M.L. and J.M Singh. Understanding Research Methodology. 1998.

Singh, Mrigendra Lal. Understanding Research Methodology. Nepal: National Book centre, 2013.