Model to determine when mean is unlikely to cross threshold

A doctor inserts a needle into a muscle to measure the duration of specific events. For every insertion approximately 5 data points are gathered. The doctor keeps making new insertions until he/she decides that enough data has been collected to put the patient in 1 of 3 boxes based on the mean of all data points.

To save time and to minimize patient discomfort, I would like to develop a tool, that informs the doctor when it is statistically unlikely that gathering more data will move the mean from one box to another.
My first thought is to calculate 95% confidence interval. If both upper and lower limit is "in" the same box, we can say it is likely not changing if the sample size is increased. But I would greatly appreciate other ideas. I´m also thinking that the closer the mean is to T1 or T2 (in the picture below) the higher the CI should be.



Less is more. Stay pure. Stay poor.
Just a general comment, why use 95% CIs. The alpha should be selected based on the level of risk you are willing to accept in regards to a type I error.

You approach doesnt set off any red flags and will be subject dependent. You are just waiting until the sample converges to the truth. Does anything change in the patient between collections or can each be considered independent. Also, what happens in the above second image if the truth is near the line? Do you indefinitely keep collecting data or call it after some many iterations??