Testing for Differences Between Two Population Means: The t-test
The t-test is a statistical test applied to evaluate whether the means of two groups are significantly different. The t-statistic, also called t-value, t-stat or simply t, may be generally described as the quotient of the difference between the means of two groups and the variability of the two groups (Harmon, 2011). There are a number of variations for the t-test. For this particular lesson, the independent samples t-test will be discussed.
In this tutorial, actual data will be utilized in demonstrating how an independent samples t-test is performed, first by computation using the formula and a scientific calculator, and then by computerized calculation using the software Statistical Package for the Social Sciences (SPSS) Version 17, and Microsoft Excel (2003). As per experience, integrating theory with a practical application works best in topics concerning numerical ability. Hence, the essentials you need to learn to be able to use independent samples t-test correctly will be elucidated as we solve our tutorial problem.
Moreover, I opted to use socially relevant statistics. If we are to spend time and effort on this tutorial, let us try shooting two birds with one stone. At the end of this tutorial, you would have learned how to properly use independent samples t-test, and you would have achieved a certain degree of awareness on issues of the day that matter. This is one aspect which will set this education site from the rest – let us not just learn statistics and research, let us learn some current events and let us be socially conscious about significant advocacies and developments all around us.
For this tutorial, we are interested in solving the following problem:
Is corruption based on the corruption perceptions index in countries with the full presidential system of government higher than countries with the parliamentary system?
Corruption is defined by Detzer (2011) as “the abuse of public power for private benefit (or profit)” (p. 2). Detzer (2011) maintained that this was the same definition adopted by the World Bank and Transparency International. Why are we interested in the aforestated problem? For one, Gerring and Thacker (2004) believe that there is some association with political corruption and the form of government. The duo argued that if polities are minimally democratic to say the least, they are bound to experience lower levels of corruption. The study conducted by Lederman, Loayza and Soares (2005) identified the relationship between a parliamentary system of government and lower level of corruption. This partly explains the interest in the research problem for this tutorial.
One issue should, however, be cleared before we push through with the statistics. How do we measure corruption? Politicians perpetrate corruption in stealth mode. In the Philippines, however, the stewards of corruption commit the act blatantly and aggressively in front of our very faces. Still, measuring corruption per se is more controversial than giving it a definition as articulated by Uslaner (2008). He even cautioned researchers that “any attempt to quantify corruption is thus fraught with danger and will endanger much criticism” (Uslaner, 2008, p. 11). This discussion will, therefore, venture to quantify corruption in terms of corruption perceptions index (CPI) as measured by Transparency International. Uslaner (2008) uses this measure of corruption, which he claims is the most widely use metric for the variable under study. For further reading on how CPI is measured, please visit the URL:
http://www.transparency.org/policy_research/surveys_indices/cpi/2010/in_detail.
http://www.transparency.org/policy_research/surveys_indices/cpi/2010/in_detail.
To solve the research problem, the following null hypothesis is formulated using a two-tailed or non-directional analysis and a 0.05 level of significance (α = 0.05):
The corruption in countries governed using the full presidential system is equal to the corruption in countries governed using the parliamentary system.
If pres is our variable for the mean CPI for countries/territories with the full presidential system of government, parl is our variable for the mean CPI for countries/territories with the parliamentary system, H0 is our abbreviation for the null hypothesis and H1 is our abbreviation for the alternative hypothesis, then our non-directional hypotheses may be stated as:
H0: pres = parl
and
H1: pres ≠ parl
The dataset for this problem was obtained from two authoritative Websites: CIA World Factbook (2011) and the Transparency International (2010). Here is our dataset:
Solution 1: Computation Using a Scientific Calculator
Since, any scientific calculator can perform descriptive statistical calculations such as computing the mean and the standard deviation, calculation for these values will not anymore be taken be taken up in this tutorial. The variables n1 and n2 are the sample size (in this case, also the population size) of the countries/territories using presidential and parliamentary systems, respectively, whose values are 41 and 43. These values are included below the dataset: xbar1 = 3.3195; s1 = 1.5179; xbar2 = 4.5977; s2 = 1.9731 The approximate normal distribution of the dataset in shown below as Figure 1.
Figure 1. Approximate normal distribution of the dataset (Note: The image was retrieved from Mehta (2011) and superimposed with the values used in this tutorial).
For the normal distribution of the dataset, the limiting values of t for a non-directional or two-tailed analysis for the null hypothesis to be accepted can be found from a t-distribution table which is available as an appendix in most statistics books or online. To find the value of the critical t, we should be aware of the level of confidence specified (α = 0.05) and the degrees of freedom (df). Df may be computed from the formula:
df = n1 + n2 – 2
df = 41 + 43 – 2
df = 82
The critical t may be obtained from the following t-distribution table. The critical t may be found by using the 0.05 two tail probabilities from column 4 corresponding to the computed df. Figure 2. Percentage Points of the t-distribution (Knight, n.d.)
It may be observed that there is no exact value for 82 degrees of freedom, but the nearest df where the critical t is available is 80, rather than 100. The critical values of t for a two-tailed or non-directional analysis are, therefore, - 1.990 and + 1.990. These values were marked in Figure 1. The area of the normal curve between – 1.990 and + 1.990 is the acceptance region. If the computed value of t falls within – 1.990 and + 1.990, the null hypothesis will be accepted. As applied to the tutorial problem, when the null hypothesis is accepted, we would have established statistical evidence that, indeed, the corruption in countries governed using the full presidential system equal to (or not statistically different from) the corruption in countries governed using the parliamentary system. The area under the normal curve to the left of - 1.990, as well as the area of the curve to the right of + 1.990 are the rejection regions. If the computed value of t falls beyond – 1.990, meaning less than – 1.990 or beyond + 1.990, meaning greater than +1.990, the null hypothesis will be rejected and the alternative hypothesis will be adopted. This connotes that the corruption in countries governed using the full presidential system is NOT equal to (or is statistically different from) the corruption in countries governed using the parliamentary system.
To calculate the t-statistic, the formula specified in Weinberg and Abramowitz (2008) is:
To calculate the t-statistic, the formula specified in Weinberg and Abramowitz (2008) is:
Substituting the values of xbar1, xbar2, s1, s2, n1 and n2 which are rounded off to four digits after the decimal point, the computed value from a scientific calculator of the t-statistic is: t = - 3.297. This point is also marked off in Figure 1 and is found to the left of the -1.990 limit. It may be observed that t = - 3.297 lies in the rejection region on the left. As earlier mentioned, this condition prompts us to reject the null hypothesis and adopt the alternative hypothesis. Therefore, the corruption in countries governed using the full presidential system is statistically different from the corruption in countries governed using the parliamentary system. By looking at their respective means, the independent samples t-test showed us that the corruption in countries governed using the full presidential system (xbar1 = 3.319) is statistically higher than the corruption in countries governed using the parliamentary system (xbar2 = 4.5977). It should be noted at this point that higher values of the corruption perceptions index indicate lower levels of corruption.
The foregoing findings offer support for the advocacy of the Constitutional Reform and Rectification for Economic Competitiveness and Transformation (CoRRECT™) Movement of which I am member of the core group. With a parliamentary system of government, we have better chances of curbing corruption in the country. The CoRRECT™ three point agenda calls for economic liberalization, evolving territorial decentralization, and shift to unicameral parliamentary system.
The next post will demonstrate the solution to the same problem using Microsoft Excel and SPSS Version 17. Tutorial videos will be provided.
References:
Gerring, J. & Thacker, S. C. (2004). Political institutions and corruption: The role of unitarism and parliamentarism, British Journal of Political Science, 34(2), 295-300.
Harmon, M. (2011). t-Tests in Excel: The complete guide. Incline Village, NV: Excel Master Series.
Knight, W. (n.d.). Percentage points of the t-distribution. Retrieved from http://www.math.unb.ca/~knight/utility/t-table.htm
Lederman, D., Loayxa, N. V. & Soares, R. R. (2005). Accountability and corruption: Political institutions matter, Economics & Politics, 17(1), 1-35.
Mehta, T. (2011). Drawing a normal curve. Retrieved from http://www.tushar-mehta.com/excel/charts/normal_distribution/
Transparency International (2010). Corruptions Perceptions Index 2010. Berlin: Transparency International Secretariat. Retrieved from http://www.transparency.org/publications/publications
/annual_reports/ annual_report_2010
Transparency International (2011). Poll results. Retrieved from http://www.transparency.org/
content/collectedinfo/31738
Uslaner, E. M. (2008). Corruption, inequality and the rule of law. New York: Cambridge University Press.
Weinberg, S. L. & Abramowitz, S. K. (2008). Statistics using SPSS (2nd ed.). New York: Cambridge University Press.
