Suppose I have range of x and y values which I have produced a linear regression model from. How would you answer this question,"Is the mean response (predicted value) for the y value at x = 12 significantly more than 16.5 at the 1% significance level"?
In my case my number of data points is 12 so a t test comes to mind, however, doesn't a t t test deal with sample means? I am not sure how to go about this hypothesis test question since this involves data points rather than sample means. I have tried using Excel to solve this question but I feel this beyond the scope of what I know.
If your model is $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, then the mean response at $x = 12$ is $\beta_0 + 12 \beta_1$. So you want to test $H_0: \{\beta_0 + 12 \beta_1 \leqslant 16.5\}$. This is called a linear hypothesis test. If you google "linear hypothesis testing" you will find some resources about these tests.