  Given ${\mathbb{F}_{p^t}}$, a field with $p^t$ elements, where $p$ is a prime
power, $t$ is a positive integer. Let $f(x)$ be a polynomial over
$\mathbb{F}_{p^t}$ of degree $m$ with some restrictions. In this paper, we
construct a sufficient condition on $(p,t)$ which guarantees the existence of a
primitive normal pair $(\epsilon,f(\epsilon))$ such that
$Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=a$,
$Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=b$ and
$N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(\epsilon)=c$,
$N_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(\epsilon))=d$ where $c,d\in\mathbb{F}_{p}$
are primitive elements and $a,b\in\mathbb{F}_{p}^*$. Furthermore, we
demonstrate that, for $p=11^k;$ $k\geq1,$ $m=8$ and $t\geq 15$, there are only
$4$ possible exceptions where such pairs may not exist.
