# Using Statistical Test Method to Establish a Decision Model of Performance Evaluation Matrix
## 5 Case Study  
As mentioned above, with the performance evaluation matrix, we can help the system administrator of the teaching APP quickly figure out the key service items that need to be improved and implement some improvements. In the field of English digital learning, many scholars have been involved in the research on the computer-assisted language learning system (CALL system) to help students use the CALL system to improve their foreign language [22; 23; 24; 25; 26]. The web-based e-learning system (WELS) satisfaction questionnaire designed by [PERSON] and [PERSON] [27], including four dimensions--1. Learner Interface, 2. Learning Community, 3. System Content, and 4. Personalization and Contains--and 13 questions can be said to be simple and complete. Therefore, it has been adopted by plenty of studies [28; 29; 30; 31], for example, on the perceptions of students' e-learning experiences in English for specific purpose (ESP) classes [28], examining and categorizing proper conditions for mathematics teaching as a real-life utilization with a multi-criteria decision analysis and fuzzy-decision-making trial and evaluation laboratory (MCDA and F-DEMATEL) method [29], and examining 98 articles to confirm the satisfaction as a critical aspect of student success in online education [30]. This paper also used this questionnaire to collect users' satisfaction and importance and adopted the Call system as a case study to illustrate the model proposed in this study. This questionnaire survey recruited students from a university in Taiwan as a sample. A total of 398 questionnaires were sent out, and 355 questionnaires were received (the response rate was 89.2%), of which 11 were invalid and 344 were effective (the effective response rate was 86.4%). The following is a data analysis based on the received 344 valid questionnaires (N = 344). First, according to Equations (3) and (14), we calculated the estimator (\(\hat{\theta}_{i}^{S}\)) of the satisfaction index and the estimator (\(\hat{\theta}_{i}^{I}\)) of the importance index of each question, respectively. At the same time, according to Equation (5), we also calculated the average of the estimators of all satisfaction indices, \(\theta_{S}\) = 0.52. Then, the hypotheses of the statistical test for satisfaction index \(i\) are displayed as follows:  
\[\text{null hypothesis }H_{0}:\;\theta_{i}^{S}\geq 0.52; \tag{40}\]  
\[\text{alternative hypothesis }H_{1}:\;\theta_{i}^{S}<0.52. \tag{41}\]  
Subsequently, based on Equation (13) and \(\theta_{S}\) =0.52, we calculated the critical value (\(C_{X0i}\)) of the danger zone of the satisfaction index of each question and filled it in Table 1.  
Similarly, according to Equation (16), the average of estimators of all the importance indices, \(\theta_{I}\) =0.70, was calculated. The hypotheses of the statistical test for importance index \(i\) were defined as follows:  
\[\text{null hypothesis }H_{0}:\;\theta_{i}^{I}\geq 0.70; \tag{42}\]  
\[\text{alternative hypothesis }H_{1}:\;\theta_{i}^{I}<0.70. \tag{43}\]Next, according to Equation (24) and \(\theta_{l}=\)0.70, we determined the critical value (\(C_{Y0i}\)) of the danger zone of the importance index of each question and filled it in Table 1.  
According to Table 1, we used horizontal line \(y=C_{Y0}\) and vertical line \(x=C_{X0}\) to divide the performance evaluation matrix into four quadrants, where \(C_{X0}=Min\big{\{}C_{X01},C_{X02},\ldots,C_{X0q}\big{\}}=0.49\) and \(C_{Y0}=Min\big{\{}C_{Y01},C_{Y02},\ldots,C_{Y0q}\big{\}}=0.68\). Next, according to the evaluation coordinate points of \((x_{i},y_{i})=\big{(}\theta_{i}^{5},\delta_{i}^{l}\big{)}\) for all service items in Table 1, we plotted them in Figure 2, as shown below. According to Figure 2, the evaluation coordinate points of these 13 service items fall into the four evaluation quadrants as follows:  
Quadrant 1: items 1-4;  
Quadrant 2: item 5, item 6;  
Quadrant 3: item 7, item 8;  
Quadrant 4: items 9-13. According to the evaluation rules, service items 5-8 need to improve.