The claim is average heating bill is less than \$ 78. In symbolic form it can be written as $\mu < \$78$.
If $\mu < \$78$ is false, then $\mu \geq \$78$ must be true.
The null hypothesis is the statement which contains equality.
Hence the null hypothesis be $H_0 : \mu \geq \$78$. And the alternative hypothesis be $H_a : \mu < \$78$.
So the hypothesis testing problem be
$H_0 : \mu \geq \$78$ against $H_a : \mu < \$78$.