With the given table, this isn't possible without resorting to an approximation of an approximation.
The interval [tex][0,8][/tex] can be partitioned as [tex][0,2]\cup[2,4]\cup[4,6]\cup[6,8][/tex], with the respective midpoints of [tex]t\in\{1,3,5,7\}[/tex]. So the average temperature is given exactly by the definite integral
[tex]\displaystyle\frac1{8-0}\int_0^8f(t)\,\mathrm dt[/tex]
which is approximated by
[tex]\displaystyle\frac14\sum_{t\in\{1,3,5,7\}}f(t)=\dfrac{f(1)+f(3)+f(5)+f(7)}4[/tex]
where the coefficient [tex]\dfrac14[/tex] comes from the fact that each subinterval has length 2, and so [tex]\dfrac28=\dfrac14[/tex].
However, you don't know the values of [tex]f(t)[/tex] at these points. At best you can approximate them, perhaps by interpolating [tex]f(t)[/tex] (you have the details needed to do it, at any rate).
Are you sure you're not supposed to find the average temperature over the entire set of observation times, i.e. over [tex]0\le t\le16[/tex]?
