# The Effect of Training in Virtual Reality on the Precision of Hand Movements
## 5 In-Depth Statistical Evaluation
### Methodology of Statistical Eevaluation  
Statistical hypothesis testing was used for the statistical processing of the experimental data. A statistical hypothesis is a statement that relates to an unknown property or parameters of the probability distribution of a random variable. In statistical testing, we always have two types of hypotheses:  
**Hypothesis 1 (H1).**: _Null hypothesis \(H_{0}\) means the hypothesis, the validity of which we verify using a test._  
**Hypothesis 2 (H2).**: _Alternative hypothesis \(H_{A}\) means the hypothesis that denies the validity of the null hypothesis._  
To test the null hypothesis, we must always choose a significance level \(\alpha\), which is the probability of rejection of a true null hypothesis (type I error). We always require this probability to be small, so we choose, according to our habit, to be \(\alpha=0.05\). The decision to reject/not-reject the null hypothesis is made using the \(p\)-value of the test. We reject the null hypothesis if the \(p\)-value of the test is less than the level of significance. More about hypothesis testing can be found in [37].  
**Statistical tests** are used to verify the validity of the null hypothesis. There are parametric and nonparametric tests. Parametric tests are those for which we must know the distribution of data, and they usually assume a normal distribution. Nonparametric tests do not require this assumption. Three tests were used in this work:  
1. Jarque-Bera test
2. Paired-sample \(t\)-test
3. Wilcoxon signed ranked test  
**Jarque-Bera test**  
First, the data obtained were tested for normality using the Jarque-Bera [38] test in order to decide whether we should use parametric or non-parametric statistical tests.  
Figure 26: One hand sine avg. time to accuracy trend.  
Parametric tests can only be used for data with normal distribution; non-parametric tests are used for data with different distributions [37].  
The Jarque-Bera test tested the null hypothesis that the measured data follow a normal distribution against the alternative hypothesis that the measured data do not follow a normal distribution.  
If the \(p\)-value is greater than the significance level \(\alpha=0.05\), we do not reject the null hypothesis of data normality and we can use a parametric test (paired-sample \(t\)-test). Otherwise, we reject the null hypothesis of data normality, and we have to use a non-parametric test (Wilcoxon signed rank test).  
**Paired-sample \(t\)-test**  
This paired-sample \(t\)-test [39] is parametric and can only be used for data that comes from a normal distribution.  
We test the null hypothesis that the distribution of the observed variable (time/distance/accuracy) is identical in both groups (first and last attempt) and has the same mean (expected value) against the alternative hypothesis that the distribution of the observed variables is not identical in both groups and has a different mean.  
We reject the null hypothesis if the \(p\)-value of the test is less than the significance level \(\alpha=0.05\), which means that there is a statistically significant difference between the first and last attempt. If we do not reject the null hypothesis, there is no statistically significant difference.  
**Wilcoxon signed rank test**  
If data does not come from a normal distribution, in the next step, we use the parametric Wilcoxon test for two paired samples [40]. It is the non-parametric equivalent of the paired-sample \(t\)-test. We test that the null hypothesis data in both groups (first and last attempt) are samples from continuous distributions with equal medians against the alternative that they are not.  
The method of evaluation and conclusions are the same as for the paired-sample \(t\)-test.