This is a shiny visualisation of CIs.
It samples from a given distribution lots and lots of times, and calculates the CI for this sample. It then visualises the CIs created so far and highlights those that do not include the mean. It also displays a cumulative histogram of the proportion of CIs that included a given range of values.
You can download it and run it inside RStudio, or click the badge at the top to run it in your browser.