Prove: Number of (not necessarily primes) not fulfilling Fermats little theorem exceed $\frac{1}{2}\varphi(n)$

Question

Show for every $n \in \mathbb{N}$:

$$ \#\{\bar a \in (\mathbb{Z}/n\mathbb{Z})^\times \,\,|\,\, \bar a^{n-1} \neq \bar1 \} \ge \frac{1}{2}\varphi(n) $$

given at least one $a \in (\mathbb{Z}/n\mathbb{Z})^\times$ with $ \bar a^{n-1} \neq \bar1.$