I am a bit confused about what is generally done (without the help of any kind of software) to obtain critical values when exact degrees of freedom values are missing in a T-table. Most T-tables tend to have degree of freedom/df values up to 100 and then jump straight to 200 and so on. But what is the best approach when suppose I have to consider df=199 where alpha/2= 0.025 in a two-tail test?
- Should I take an average of t0.025,100 and t0.025,200?
- Should I only consider t0.025,100 to get the critical values?
When you don't see the exact degrees of freedom in the table, then the most appropriate thing to do is apply a linear interpolation to adjacent table entries. For instance, the critical value for a two-sided test with $\alpha = 0.0001$ corresponds to the last column in the table. If the degrees of freedom is $\nu = 32$, we only have the critical values for the following:
$$\begin{array}{c|c} \nu & t_{\nu}^* \\ \hline 30 & 4.48 \\ 35 & 4.39 \\ \end{array}$$
So linear interpolation for $\nu = 32$ would look like this: $$t_{32}^* \approx \frac{35 - 32}{35 - 30} t_{30}^* + \frac{32 - 30}{35 - 30} t_{35}^* = \frac{3}{5} (4.48) + \frac{2}{5} (4.39) \approx 4.44.$$
In fact, the exact value is $4.44105393811587\ldots$. So it's not too bad. However, interpolation is not advisable if the table values are too far apart.