Average impact of multiple conversion errors
APL (Dyalog Unicode), 41 bytes

(|40+⎕)+.×(|÷≢)1-1.8*1(-,+)⍨⍣(⎕-2)⊢1-3|⍳4

Try it online!
A full program. Takes the number of people, then the initial temperature. TIO version is wrapped in a function for easier testing. As I pointed out in a comment, the initial and final units do not matter as long as they are equal.
The math
The code exploits the fact that Fahrenheit-Celsius conversion has a fixpoint at -40 (-40°F = -40°C). Then we can add 40 to all temperature readings, and then the whole series of conversion reduces into a (positive or negative) integer power of \$\frac95\$.
$$
F+40=\frac95(C+40), \quad C+40 = \frac59(F+40)
$$
Since every temperature value is offset by the constant value of 40, the difference between two offset temperature values is the same as that of the original. Therefore, it suffices to apply the offset to the initial temperature and ignore it to the end.
Let's call \$r = \frac95\$.
Now to the part of counting all conversion paths. Let's say the initial reading is in Fahrenheit. Then:

Initial conversion: F→F (\$r^0\$) or F→C (\$r^{-1}\$).
Final conversion: F→F (\$r^0\$) or C→F (\$r^1\$).
All inner conversions: C→F (\$r^1\$) or F→C (\$r^{-1}\$).

So the task reduces to finding all the individual terms in the expansion of \$(r^0 + r^{-1})(r^1+r^{-1})^{n-2}(r^1+r^0)\$. Then plug in the value of \$r\$, evaluate the average of differences to 1, and then multiply with the initial temperature (with offset).
The code
(|40+⎕)+.×(|÷≢)1-1.8*1(-,+)⍨⍣(⎕-2)⊢1-3|⍳4

⊢1-3|⍳4  ⍝ A fancy way to create an array of 0 ¯1 1 0
         ⍝ which represents the powers in (1+r^-1)(r+1)
⍣(⎕-2)   ⍝ Repeat n-2 times...
1(-,+)⍨  ⍝ multiply the given polynomial with (r+r^-1)
         ⍝ and list all powers
1.8*     ⍝ Plug in r = 1.8 and evaluate the powers
1-       ⍝ Difference of each value from 1
(|÷≢)    ⍝ Average of absolute differences from 1
+.×      ⍝ Product and sum with...
(|40+⎕)  ⍝ Absolute difference between -40 and t