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.


Prediction Intervals
Beyond the second insertion, you may be able to make a 95% prediction interval for what the next insertion will show. The appropriate endpoint of that interval will help you to know if the new value will likely lead to a change in the 'box'.
Suppose you have independent normal observations $X_1, X_2, \dots, X_n$ with mean $\bar X_n$ and standard deviation $S_n.$ Then a 95% prediction interval for
new observation $X_{n+1}$ from the same distribution is of the form
$$\bar X_n \pm t^*_{n-1}S_n\sqrt{1 + \frac 1 n},$$ where $t^*_n$ cuts probability $0.025$ from the upper tail of Student's t distribution with $\nu = n-1$ degrees of freedom.
Contrast this 95% PI with the more familiar (and shorter) 95% CI $\bar X\pm t_{n-1}^*S_n/\sqrt{n}$ for the unknown mean $\mu.$ For the derivation,
see Wikipedia on prediction intervals with unknown mean and variance.
