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Inferential Statistics Article Critique


An analyst is faced with a large population in any study. Collecting a sample gives a small representative of the entire population. One may use descriptive statistics to compute the mean and standard deviations of the data. However, methods such as the chi-square test of goodness fit, the chi-square test of independence, and the t-test enable one to establish the significance of the variables. A two-tailed t-test gives the relationship between mental health and exercise. An article critique of a sample of players and non-players provides insight into applying a t-test.


The authors were trying to establish the effect of physical exercise on mental health. They took a sample of 40 male individuals, half being sports people and the others non-sport persons from Amravati City. The former is active daily, whereas the others don’t participate in strenuous activities. The age range was between 18 and 28 years. The question being asked is whether exercise contributes to better mental health. The hypothesis being tested is players have better mental health than non-players. The test used was the t-score. It considered the mean, the standard deviation, the mean deviation, and the calculated t-value two-tailed test. The t-test was an ideal method since it minimizes the standard error in a sample (Tanner, 2016). The analyst applied an alpha of 0.05 in establishing the critical value. A comparative analysis of the t values and the critical value shows that the latter exceeded the former in all the variables except life satisfaction. One notes that life satisfaction occurs by chance. The choice of six variables is enough. However, the non-players and the players have different means. It’s funny that each variable has similar t-values for these two groups. For example, the players have a mean of 40.65 for anxiety, whereas the non-players have 46.35. It is funny that the variable has a t-value of 3.91.


Table 1 gives the t-values of the six variables, namely: Loss of behavior, anxiety, emotional ties, positive affect, depreciation, and life satisfaction. It is established that the critical value was at an alpha of 0.05, and a degree of freedom of 38 is 2.0168. One compares the t-values with the essential values (Tanner, 2016). It is established that the variables, except life satisfaction, have a t-value greater than the critical value. As a result, these variables significantly differ between the players and non-players. Conversely, I think the analyst should have divided the players and non-players into two subdivisions—those with mental health issues and those without mental health issues. Having a sample with many individuals with depression, whether they are players or non-players, may skew the distribution, resulting in different results. Besides, I will improve the study by increasing the number of players in the sample. Using 100 players and 100 non-players will likely reduce the errors inherent in a small sample.


The t-test introduces the standard error, hence an improvement from the chi-square tests. It shows the significance of each variable with mental health. One easily gets the significant ones and those that are not. However, the chi-square test of independence is an ideal method in this case compared to the t-test. A contingency table is drawn to depict all the data collected in the test (Tanner, 2016). There is a need to have two columns of players and non-players that give the frequencies of the various mental issues. The rows consist of anxiety, depression, loss of behavior, positive affect, emotional ties, and life satisfaction. This technique considers the relationship between mental health and exercise. The six variables are enough for this study. The (r-1)(k-1) would enable me to determine the degree of freedom and hence find the critical value (McHugh, 2013). It is noted that there is only one z-distribution. However, there are many t-distributions. As a result, the t-test has critical values for each variable based on whether one performed a one-tailed or two-tailed. This situation makes the study tedious since one has to compare the t-values and the critical ones to establish significance. The degree of freedom also depends on whether the evaluation is a one-sample or two-sample t-test. The latter has a degree of freedom of n-1, whereas the latter has n-2. As a result, the significance is based on the type of test and the degree of freedom.


The t-test assumes the homogeneity of variance. It is believed that both the players and non-players have similar standard errors. This situation may need to be more practical and realistic. It also assumes a t-distribution in the difference between the two means. However, this situation is only practicable for small samples. The method is also ideal for comparing only two groups. Having more than two variables results in a type 1 error.


The t-test is essential in evaluating the variables in two groups. For example, a psychologist may determine the poverty levels and mental health. Such a study has the poor and the wealthy participants. It then considers variables such as depression, suicide levels, and stress levels. Besides, one may evaluate whether the treatment methods have a corrective action on the patients. This move is ideal for therapy in the rehabilitation of victims of sexual assault or substance abuse.


T-test is an ideal method in establishing the significance of various on an issue. The technique is an improvement on the z-score by considering the standard error. It is believed that physical exercises improve the mental health of the respondents. Testing such a hypothesis requires one to establish the variables in mental health. One determines the mean, the standard deviation, and the standard error. The t-value is computed and compared with the critical value. Such a method enables one to determine the one or two-tailed test.


McHugh, M. L. (2013). The chi-square test of independence Links to an external site. Biochemia Medica, 23(2), 143–149.

Tanner, D. (2016). Statistics for the behavioral & social sciences (2nd ed.). Bridgepoint Education.


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