In many types of research, a scientist might be tasked to empirically determine whether there is a significant difference in the means of two distinct populations with respect to a certain variable. Probably the study would require the researcher to proof beyond reasonable that the two population differ significantly in yielding a certain variable of interest. If the study is of two plants, then the researcher is tasked to show which of the two plants will have a higher yield in the climatic condition of a region and recommend it for use by farmers. The study could also be a two transport processes designed to reduce accidents on a road. The process that has a statistically significant lower mean can be adopted to control traffic on the particular road.
In the country, income distribution among races is a matter of public concern. The government has often claimed to be doing everything in its powers to harmonize the income of the various races in the country. However, have these effort helped? Is the deviation of income among the blacks and white statistically insignificant? These are a valid question that social scientist seek to answer across all the spheres of private and public employment. Therefore, a scientist will endeavour to statistically significant difference in the mean income of white and black in the country (Ghasemi & Zahediasl, 2012).
The variables of interest for the research is total family income and the race of the participant. A participant would be asked to state his/her race and state his average income. The two races of interest are blacks and whites. A proper null hypothesis for the research would be that the total family income for the two races is same while an appropriate alternative hypothesis would be that there is an insignificant difference in the total family income among the two races. The determination of the race of an individual would be easy as its a simple categorical variable. The skin tone of an individual will qualify him/her to be categorised in a particular race. The total family income will be measured based on the amount of income that the participant makes, plus the contribution of his/her spouse or any other adult listed in the respondents tax form. The nature of this variable is quantitative. The race of the respondent will be our dependent variable while the total family income of the participant will be an independent variable ("Independent T-Test - An introduction to when to use this test and what are the variables required | Laerd Statistics", 2016).
The sample of respondents will be determined by simple random sampling so as to obey the principles of randomization and error or local control. Obeying the two principles will improve the internal validity of the data and increase the precision of the results. Many statistical tools will require that data be normally distributed if this important assumption is violated many statistical tests to determine hidden elements of the data will fail. The concept of the law of large numbers indicates that the larger the sample size, the more likely the sample statistic will approach the population parameter, and thus the more likely the data will normally be distributed. The sample since it will be randomly selected and appropriately large we also expect it to be normally distributed and therefore allow the research to employ an Independent t-test to test the null hypothesis.
In conclusion, to use the test, the other assumption other than normality that needs to be tested. The study should determine whether the two populations being compared have the same variance. The study should also ensure that the samples obtained from the two population are sampled independently and have a normal distribution with a mean m and variance d^2.
Ghasemi, A. & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide for Non-Statisticians. Int J Endocrinol Metab, 10(2), 486-489. http://dx.doi.org/10.5812/ijem.3505
Independent T-Test - An introduction to when to use this test and what are the variables required | Laerd Statistics. (2016). Statistics.laerd.com. Retrieved 1 May 2016, from https://statistics.laerd.com/statistical-guides/independent-t-test-statistical-guide.php
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