Suppose task performance $y$ increases with trial number $x$ ($x = 1, 2, …, 10$), so that there is a practice effect. Let’s suppose the practice effect is linear. Subjects have a long break, and then provide another observation of $y$ (for $x = 11$).
I have reason to believe that subjects get better at the task during this break, beyond what would be expected by the linear practice effect. Therefore, I want to test whether average performance on the 11th trial is predicted by a linear model fit to the first 10 trials. What would be the correct procedure to test this hypothesis? I imagine it would be something like
- Fit a regression model of y on x for x = 1,2,…,10.
- Compute the conditional mean given $x = 11$ based on the model and call this $Y_m$.
- Compute the sample mean of $y$ for $x = 11$ and call this $Y_s$.
- Test for a difference in the above two quantities.
But how do I compute the standard error of the difference? Or should I just compute confidence intervals for both $Y_m$ and $Y_s$ and see if they overlap?
Update: Can I fit a linear regression to all trials $x = 1,2,...,11$, include a dummy variable for the 11th trial, and test the significance of the dummy variable term?