Pratt and Cullen noted that the computation of the t-test statistic provides a measure of the equality of the hypothesized mean and the statistical mean, which is expressed as the standard deviation of the difference between the two mean values that occur as between-subjects and within-subjects (3). A statistical analysis of the case study in table 1, table 2, and table 3 show the computations performed to determine the significance of quiz 1 and quiz 2 scores based on a test of normality and homogeneity on a sample size of 105 participants (Coladarci, Cobb, Minium and Clarke, 24). The test of normality that was based on Kolmogorov-Smirnov^{a }is a clear indication of the statistical measures of the significance of the GPA scores.

*Table 1: One-Sample Statistics.*

There were no significant differences in the responses between the standard deviation for the GPA scores in the test statistic for the mean GPA score and the gender of the participants. However, the tests are just samples of the entire tests that were done on the results for both quizzes 1 and 2.

*Table 2: Percentiles.*

Table 2 shows the weighted averages between the male and female participants and mean GPA scores for the grades, A, B, C, D, E., and F.

*Table 3: Correlations.*

Often, t-statistic values are not normally distributed. A large population or a big n value generates smaller critical values while small n value generates big critical values (Rodgers 12). When affirming or rejecting the null hypothesis, it is not necessary to get a big *t*-score to reject the null hypothesis showing that the power of the t-test increases as the sample size increases and its accuracy depends on the degree of freedom (df).

## Application

Rodgers notes that the t-statistic can be applied to conduct a binomial test to determine the proportion of success of the two-level categorical dependent variable, Chi-square goodness of fit to determine the deviation of the categorical variable to hypothesized proportions based on a one-sample t-test. Two independent samples t-test compares the mean of normally distributed variables. The test of normality is done to determine the significance of the two-level categorical dependent variable using a 95% significance test.

## Strengths and limitations of various statistical tests

A one-sample t-test on 95% significance compares the sample mean with a hypothesized sample mean. The results are easy to interpret. The t-test statistic has significance and value in meaning if the sample size and the difference between the mean are large. According to Washington, Karlaftis, and Mannering, the responses must be consistent for the results to be regarded as having significance (3). Enders notes that the t-test only examines the mean without any reference to the individual scores leading to conclusions that are based on the mean and not individual performance as is evident in the case study (4).

## Strengths and limitations of an independent samples t-test

The independent samples t-test could enable the study to verify the claim that ethnicity for either the male or female participants affects their GPA scores. This implies that the responses from two groups, male and female can be compared. Also, it ensures that the samples are equivalent and can be matched on a one-on-one basis, the sample size used in the could be small, less variability in scores allows for repeated measures that reduce the random error is possible and enables good control of individual differences. However, repeat measures are susceptible to order effects and are likely to suffer the carry-over effects.

## Decision-making process

The accuracy and reliability of the output statistic values show reliability depending on the type of test to be conducted, the data to use, and the source and reliability of data. A wrong decision or incorrect hypothesis could lead to either type I or type II errors. The decision to accept or reject the statistic results depends on t-statistic values.

## Assumptions

When conducting the statistical analysis, it was common to assume that the GPA scores of each participant were independent (King, Rosopa and Minium 20). However, the test could be violated if each participant had repeated scores. In this case, the population was assumed to be normally distributed and the variances of the scores were the same.

## Hypothesis testing

Based on the research question on: What are the difference in the mean GPA score of females and males?

- H
_{0}(Null hypothesis) =4: (There is a mean difference in mean GPA among the female and male participants) - H
_{1}(Alternative hypothesis) ≠ 4: (There is no mean difference in mean GPA among the female and male participants)

*Table 4: One-Sample Test.*

The mean difference in the GPA scores between the female and male participants is -16.383, which is an extreme value on a 95% confidence interval. The degree of freedom (df) is 104 and the statistical significance of the one-sample t-test.is on a Sig. (2-tailed) in this case is of *p* <.05 (*p* =.000).

## Conclusion

In conclusion, the null hypothesis is rejected in support of the alternative hypothesis because the t-static is large (-16.383), showing that the probability of making the hypothesis is very low. That shows strong evidence against the null hypothesis because the observed effect might not have arisen purely by chance. In theory, the results are said to be statistically significant. In conclusion, the results are in support of the alternative hypothesis showing that there is no difference in the mean GPA score between male and female participants. Despite the results refuting the null hypothesis, when interpreting the results, it is necessary to avoid making type I and type II errors respectively.

## Works Cited

Coladarci, Theodore, et al. *Fundamentals of statistical reasoning in education*. John Wiley & Sons, 2010. Print

Enders, Craig K. *Applied missing data analysis*. Guilford Press, 2010. Print

King, Bruce M., Patrick J. Rosopa, and Edward W. Minium. *Statistical reasoning in the behavioral sciences*. Wiley Global Education, 2010. Print

Pratt, Travis C., and Francis T. Cullen. “The empirical status of Gottfredson and Hirschi’s general theory of crime: A meta‐analysis.” *Criminology* 38.3 (2000): 931-964.Print.

Rodgers, Joseph Lee. “The epistemology of mathematical and statistical modeling: a quiet methodological revolution.” *American Psychologist* 65.1 (2010): 1. Print

Washington, Simon P., Matthew G. Karlaftis, and Fred L. Mannering. *Statistical and econometric methods for transportation data analysis*. CRC Press, 2010. Print