# Using Statistical Test Method to Establish a Decision Model of Performance Evaluation Matrix
###### Abstract
In addition, many studies assert that since Beta assigns values between 0 and 1, it represents satisfaction ranging from 0, which is completely dissatisfied, to 1, which is 100% satisfied. Similarly, the importance ranges from 0, which means total ignorance, to 1, which refers to great importance. Consequently, satisfaction and importance are regarded as random variables subject to the Beta distribution [12; 13]. [PERSON], [PERSON], and [PERSON] [14] proposed a standardized and significant importance index and satisfaction index based on the two parameters assigned by Beta distribution. If random variable \(X_{i}\) represents the satisfaction of the \(i\)th service item, then \(X_{i}\) is distributed as the Beta distribution denoted by \(X_{i}\sim Beta\left(a_{i}^{S},b_{i}^{S}\right)\), where \(a_{i}^{S}\) and \(b_{i}^{S}\) are two parameters. Similarly, if random variable \(Y_{i}\) indicates the importance attached to the \(i\)th service item, then \(Y_{i}\) is distributed as the Beta distribution denoted by \(Y_{i}\sim Beta\left(a_{i}^{I},b_{i}^{I}\right)\), where \(a_{i}^{I}\) and \(b_{i}^{I}\) are two parameters. Thus, the satisfaction index and the importance index can be expressed, as follows, for satisfaction:  
\[\theta_{i}^{S}=\frac{a_{i}^{S}}{a_{i}^{S}+b_{i}^{S}}\;(\text{satisfactionindex}), \tag{1}\]  
\[\theta_{i}^{I}=\frac{a_{i}^{I}}{a_{i}^{I}+b_{i}^{I}}\;(\text{importanceindex}). \tag{2}\]  
According to the abovementioned data, the values of the two indices are between 0 and 1. In this paper, we set the satisfaction index as the \(x\)-axis and set the importance index as the \(y\)-axis to form the performance evaluation matrix. Since the indices have unknown parameters, if the evaluation is performed directly by the point estimates of the indices, there will be a risk of misjudgment due to sampling error [15; 16; 17]. In addition, most studies just determine the critical-to-quality (CTQ) that needs to be improved, without discussing the processing rules in resource-limited settings, and without the confirmation of the improvements. In order to make up for such research gaps, this paper adopts the one-tailed statistical hypothesis testing method of satisfaction index, determines the CTQ service items that need to be improved, and then defines their improvement order. Then, this paper uses a statistical two-tailed hypothesis test to verify whether the satisfaction of all CTQ service items that need to be improved is improved. First, this paper detects the unbiased estimators of the satisfaction index and the importance index for all service items. However, according to numerous studies, if the PEM locations of the point estimates are directly used to determine whether the service items need to be improved, then the risk of misjudgment may be caused by sampling error [18; 19]. Therefore, this paper uses the average value \(\theta_{0}^{S}\) of all satisfaction indices as the testing standard in accordance with the spirit of continuous improvement on the comprehensive quality management and employs the statistical one-tailed test to determine the critical value \(C_{0}^{S}\). At the same time, the average value \(\theta_{0}^{I}\) of all importance indices is adopted as the testing standard, and the statistical one-tailed test is employed to determine the critical value. Next, this paper establishes the evaluation blocks and evaluation rules of PEM based on the critical value derived from the statistical test and judges whether the service item needs to be improved according to the location of the evaluation coordinate point \((x_{i},y_{i})=\left(\hat{\theta}_{i}^{S},\hat{\theta}_{i}^{I}\right)\) of each service item, where \(\hat{\theta}_{i}^{S}\) is the estimator of \(\theta_{i}^{S}\) and \(\hat{\theta}_{i}^{I}\) is the estimator of \(\theta_{i}^{I}\). Meanwhile, according to the locations of the evaluation coordinate points of the service items that need to be improved in the evaluation quadrant of PEM, the priority of improvement is determined under the condition of limited resources.