Define relationship between confidence intervals and significance tests?

Test:
Denote ${\displaystyle \tau }$ as a population parameter of interest. Suppose sample data are collected from the concerned population, and the sample statistic, T, considered as a suitable estimator of ${\displaystyle \tau }$, and it must be tested whether the sample actually belongs to the population with parameter ${\displaystyle \tau }$, based on the sample data and the value of T.
A suitable test of significance helps to determine whether the value of T suggests that the data truly are from the population with parameter ${\displaystyle \tau }$.
The significance level, ${\displaystyle \alpha }$ is usually mentioned for the test, and the conclusion is arrived at, with respect to the value of ${\displaystyle \alpha }$.
The p-value may be used to reach at the conclusion.
Decision rule using p-value: Reject H0 at significance level ${\displaystyle \alpha }$, if p-value ${\displaystyle \le \alpha }$. Otherwise, fail to reject H0.
Step 2Confidence interval:
For given significance level, ${\displaystyle \alpha }$, the corresponding confidence level is (1 – ${\displaystyle \alpha }$). By convention, the confidence level is expressed as a percentage, rather than a proportion.
The 100 (1 – ${\displaystyle \alpha }$) % confidence interval for the population parameter, ${\displaystyle \tau }$, gives an interval, such that 100 (${\displaystyle \alpha }$/2) % of the observations lie below the lower limit of the confidence interval, and 100 (${\displaystyle \alpha }$/2) % of the observations lie above the upper limit of the confidence interval; 100 (1 – ${\displaystyle \alpha }$) % of the observations lie within the lower and upper limits of the confidence interval.
One can say with 100 (1 – ${\displaystyle \alpha }$) % confidence that the parameter ${\displaystyle \tau }$ lies somewhere between the lower and upper limits of the confidence interval. In other words, if several samples of the same size (or comparable sizes) are taken repeatedly from the population and the 100 (1 – ${\displaystyle \alpha }$) % confidence intervals are calculated for each sample, then about 95% of those confidence intervals will contain the true value of ${\displaystyle \tau }$.
Conclusion:
The conclusions from the confidence interval at the 100 (1 – ${\displaystyle \alpha }$) % confidence level and the significance test at significance level, ${\displaystyle \alpha }$ usually point to the same thing.
If a test is significant at significance level, ${\displaystyle \alpha }$, then the 100 (1 – ${\displaystyle \alpha }$) % confidence interval would usually not contain the value of ${\displaystyle \tau }$ as suggested in the null hypothesis.
Conversely, if a test is not significant at significance level, ${\displaystyle \alpha }$, then the 100 (1 – ${\displaystyle \alpha }$) % confidence interval would usually be highly likely to contain the value of ${\displaystyle \tau }$ as suggested in the null hypothesis.
While a point estimate is used to test the significance, the confidence interval uses the point estimate and a suitable 100 (1 – ${\displaystyle \alpha }$) % confidence level to give an interval estimate of the parameter ${\displaystyle \tau }$, with information on how likely the interval is to hold the true parameter.