## 5 Discussion  
### \(t\)-Test Statistical Significance of the Coefficients of Regression  
In regression analysis, the \(t\)-test is commonly used to assess the statistical significance of individual regression coefficients. This test helps determine whether the independent variables (e.g., gas prices) have a statistically significant effect on the dependent variable (e.g., gas consumption). Let us consider the following regression model:  
\[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\varepsilon, \tag{1}\]  
where \(x_{1}\) is \"the natural gas cost of 1 m\({}^{3}\)\" (defined as a sum of price and tariff on distribution), \(x_{2}\) is \"the average monthly temperature\", \(y\) is dependent/response variable that denotes \"the volume of gas consumption per month\", \(\beta_{0}\) is \(y\)-intercept (constant term), \(\beta_{1},\beta_{2}\) are slope coefficients for each independent/predictor variable and \(\varepsilon\) is the model's error term. Whenever we perform linear regression, we want to know if there is a statistically significant relationship between the independent/predictor variable and the dependent/response variable. We test for the significance of coefficients in the regression model by performing a \(t\)-test for the regression slope. We use the following null and alternative hypotheses for this \(t\)-test:  
Null hypothesis \(H_{0}:\ \beta_{1}=0\). Alternative hypothesis \(H_{A}:\ \beta_{1}\
eq 0\). The null hypothesis states that there is no relationship between the predictor variable and the response variable. The test statistic is as follows:  
\[t=\frac{\hat{\beta}_{1}}{\sigma_{\hat{\beta}_{1}}}, \tag{2}\]  
where \(\sigma_{\hat{\beta}_{1}}\) is the standard error of \(\hat{\beta}_{1}\). The relevant distribution is the \(t\)-distribution, with degrees of freedom \(n-(k+1)\), where \(k\) is the number of explanatory/predictor variables. Using the calculated \(t\)-statistic and the degrees of freedom, we can find the \(p\)-value from the \(t\)-distribution. The \(p\)-value indicates the probability of observing a \(t\)-statistic as extreme as, or more extreme than, the one calculated if the null hypothesis is true. Compare the \(p\)-value to chosen significance level \(\alpha\). If the \(p\)-value is less than \(\alpha\), we reject the null hypothesis and conclude that the coefficient is statistically significant. Let us consider the following data for a single consumer (Table 1) and apply the model described above. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time Period** & **Consumption,** & **Temperature,** & **Gas Cost of 1 m3,** \\  & **m3** & \({}^{\circ}\)C & **UAH** \\ \hline
2019-06 & 18 & 21.7 & 8.732320 \\ \hline
2019-07 & 24 & 18.9 & 7.9350  
F-statistic: a statistic for testing the overall significance of the model. We decide on the significance level \(\alpha=0.05\). Here, a t-value of \(\beta_{1}\) equals \(-0.239\), \(p-\text{value}=0.812\) and \(p-\text{value}>\alpha\), indicating that the coefficient is not statistically significant. This means there is sufficient evidence to suggest that the coefficient \(\beta_{1}\) equals zero. The null hypothesis thus cannot be rejected. Using the described model to the data of each consumer, we found there is no relationship between the predictor variable (\"the natural gas cost of 1 m\({}^{3\cdot\cdot}\)\") and the response variable (\"the volume of gas consumption per month\") in 90.77% of cases (20,673 cases above and equal and 2101 cases under the significance level respectively). Residuals:  Min IQ Median 3Q Max -74.516 -10.931 4.033 13.697 80.546 Coefficients:  Estimate Std. Error t value Pr(>|t|)  (Intercept) 99.3617 19.1851 5.179 6.68e-06 ***  temperature -4.2003 0.5019 -8.369 2.52e-10 ***  cost -0.4897 2.0463 -0.239 0.812 --- Signif. codes: 0 '***' 0.001 '***' 0.01 '*' 0.05 '.'