When we use Excel’s T.INV.2T() function, we input the area outside of the confidence interval. This area is also called α (alpha). See figure 49.1 below to better understand this area.

We can see from the above graph that α (alpha) makes up the area in the upper and lower tails (split between the two tails). It can be calculated using:

$$\alpha =100\mathrm{\%}–\text{Confidence Level}=1–\text{Confidence Level}$$

We can solve for $t$ using:

$$t=\text{T.INV.2T}(\alpha ,df)$$

Finally, remember that $df=n-1$

When we use Excel’s T.INV() Function, we input the area to the left of a $t$-score. The easiest is to input the area in the left tail $=\frac{\alpha }{2}$. This returns the negative (−) $t$-score:

We can see from the above graph that we can solve for the negative (−) or positive $t$-scores:

$$-t=\text{T.INV}(\frac{\alpha }{2},df)=\text{T.INV}(\frac{\alpha }{2},n-1)$$

$$t=\text{T.INV}(\frac{\alpha }{2}+\text{Conf Level},df)=\text{T.INV}(\frac{\alpha }{2}+\text{Conf Level},n-1)$$

When we use Excel’s CONFIDENCE.T() function, we input the area outside of the confidence interval (α). This function returns the Margin of Error (E):

Excel’s CONFIDENCE.T() function is the quickest way to calculate the margin of error (there is no need to calculate the $t$-score nor do the margin of error calculation):

$E=t\cdot \frac{s}{\sqrt{n}}=\text{CONFIDENCE.T}(\alpha ,2,s,n)$

We can easily calculate the lower and upper limits once the margin of error is known:

$$C{L}_{Lower}=\overline{x}-E$$

$$C{L}_{Upper}=\overline{x}+E$$