Paired t-test

Paired t-test.

In paired t-test, we, consider the method for testing of hypothesis for two samples that are dependent, which means that they are paired or matched.

Dependent sample

If one sample is related to the other the samples are dependent. Such samples are often referred to as paired samples or matched samples (because we get two values from each subject, or we get one value from each of two subjects sharing the same characteristic.

Independent sample.

Two sample are independent if the sample select from one population is not related to the samples selected from the other population.

A paired t-test can be very effective when individuals show lots of variations from one to another. Since it concentrates on charges, it can ignore the variation in absolute levels of the individuals.

Again, some assumptions are required for the validity of the paired t-test, the first assumption is that the elementary units being measured are a random sample selected from the population of interest. Each elementary unit produces two measurements. Next, look at the data set consisting of the differences between these sets of measurements. The second assumptions are that these differences are normally distributed.

Analysis steps for applying paired t-test.

Step 1. Null hypothesis (H₀):µ_x=,µ_y ie there is no significance difference in the observation before and after treatment or, the treatment is not effective .

Step 2. Alternative hypothesis (H₁).

1. H1: :µ_x≠,µ_yI,e there is significance difference in the observation before and after treatment.
2. H1::µ_x>,µ_y (right-tailed test) I,e there is positive (negative) impact in the observation after treatment).
3. H1::µ_x=,µ_y (left tail test) I,e there is negative (positive) impact in the observation after treatment.

Step 3. Test statistic.

Under H₀, the test statistic is.

$$\frac{\overline{d}}{\frac{sd}{\sqrt{n}}}$$

$$Where\,\overline{d}=Mean\,of\,the\,difference=\frac{∑d}{n}$$

$$d=X-Y=Y-X=difference\,between\,two\,pair\,of\,observation$$

$$S_d^2=Sample\,variance\,of\,the\,difference$$

$$=\frac{1}{n-1}∑(d-\overline{d})^2=\frac{1}{n-1}[∑d^2-\frac{(∑d)^2}{n}]$$

Step 4. The level of significance.

We use level of significance (α)=5% unless otherwise stated and specify whether the alternative hypothesis is one tailed or two tailed test.

Step 5 Degree of freedom:n-1

Where n= a number of the sample pair of observation.

Step 6. Critical value.

The tabulated or critical value of t at (α) % level of significance for the (n-1) degree of freedom in a one-two tailed test is obtained from t=tables.

Step7:Decision

If calculated it is less than or equal to a tabulated value of t , it falls in the acceptance region and the null hypothesis is accepted and if calculated it is greater than tabulatedΙ t_α(n-1)ιHo is rejected at the adopted level of significance.

F-distribution

F-distribution is defined as a ratio of two independent chi- square varieties divided by their respective degree of freedoms.

If the random variable x₁ is a (Χ₁²) with a v1 degree of freedom and x₂ an independent Χ²₂ variate with a v₂ degree of freedom , then by definition f-statistic is as follows.

$$F=\frac{Χ1^2}{V1}\frac{Χ_2^2}{V_2}˜F(u_1,u_2)if\,u_i>u_2$$

$$Where\, v_1=n_1-1and\,v_2=n_2-1$$

$$n_2= Number\, of \,observation \,of \,the\, first\, sample$$

$$n_3= Number\, of\, observation \,of \,the \,second \,sample$$

Assumption in f-test

The basic assumption of F- test is as follow:

• The population for each sample must be normally distributed with identical mean and variance.
• All sample observations are based on random selection and are independent.
• The ratio σ₁² to σ²₂ should be equal to or greater than 1 because the larger variance is divided by the smaller variance.
• The f – distribution is always formed by a ratio of variances . so , it is always positive.
• The total variance of the various sources of variance should be additive.

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.