# Design and Research of a Virtual Laboratory for Coding Theory
## 3 Results
### Analysis of the Results of the Student Success Rate
#### 3.2.1 Statistical Hypothesis Checking of the Initial State of the Two Groups
The decision criterion is as follows:  
- If \(\alpha\leq\alpha_{s}\), the hypothesis \(\mathbf{H_{0}}\) is accepted (there is no statistically significant difference \(\overline{X_{c}}=\overline{X_{c}}\));  
- If \(\alpha>\alpha_{s}\), the hypothesis \(\mathbf{H_{1}}\) is accepted (there is a statistically significant difference \(\overline{X_{c}}\
eq\ \overline{X_{c}}\)). The test results are given in Table 4.  
* **Checking for equality of variances** To ensure the validity of the \(t\)-test results, it is necessary to check for equality of variances of the two compared groups. This test can be performed using _Levine's Test_, which is used to test the hypothesis of equality of variances of two or more samples. It is a statistical test that checks whether the sample variances are equal or not [31]. From _[PERSON]'s test_ (Table 4--the first part), it follows that the level of significance \(\alpha_{s}\) (denoted by \(Sig\)) is greater than the predetermined risk of significance \(\alpha\):  
\[\alpha_{s}=0.112\ >\ \alpha=0.05, \tag{1}\]  
From which it follows that the means of the two groups **have approximately equal variance**. - Test for statistically significant difference  
Since the two groups have the same variances, the line _equal variances assumed_ from Table 4 is considered. There it can be seen that for the level of significance _Sig. 2-tailed_ it is obtained:  
\[\alpha_{s}=0.457\ >\ \alpha=0.05. \tag{2}\]  
After this check, the hypothesis \(H_{0}\) should be accepted, which states that there is no statistically significant difference between the two groups. This proves that there is an equal start of the control and experimental groups at the beginning of the experiment. The analysis of the statistical test for difference in means can be extended by evaluating the so-called \"effect size\". Determining the effect size allows one to assess the practical significance of how large or small the difference in mean success rates is between the two groups, not just whether the difference is statistically significant. One way to calculate effect size is through Cohen's Coefficient (Cohen's d), which is defined as follows [27; 32]:  
\[d=\frac{M_{1}-M_{2}}{SD_{pool}}. \tag{3}\]  
where  
* \(M_{1}\) and \(M_{2}\) are the mean values of average grades for the control and experimental groups, respectively, where \(M_{1}=4.5829\) and \(M_{2}=4.6810\) (Table 1). * \(SD_{pool}\) is the average standard deviation of the two groups, which is calculated as follows: \[SD_{pool}=\sqrt{\frac{(n_{1}-1).SD_{1}^{2}+(n_{2}-1).SD_{2}^{2}}{n_{1}+n_{2}-2 }}.\] (4) where
* \(n_{1}\) and \(n_{2}\) are, respectively, the sizes of the samples, which in the specific case for both groups are \(n_{1}=n_{2}=59\) (Tables 2 and 3);
* \(SD_{1}\) and \(SD_{2}\) are the standard deviations for the control and experimental groups, respectively, where \(SD_{1}=0.77649\) and \(SD_{2}=0.64586\) (Table 1). After substitution with the corresponding values \(SD_{pool}\) is obtained:  
\[SD_{pool}=0.71417 \tag{5}\]  
and [PERSON]'s d is obtained:  
\[d=-0.13736 \tag{6}\]  
When calculating [PERSON]'s d, the value of \(d=-0.13736\) means that there is a small difference between the two groups, with the \"-\" sign indicating that the mean of the control group is slightly lower than that of the experimental group. However, the effect value is quite small.