Answer
$-z_{\frac{α}{2}}\lt z_0\lt z_{\frac{α}{2}}$: null hypothesis is not rejected.
There is not enough evidence to conclude that the stock price does not fluctuate randomly.
Work Step by Step
$H_0:~The~sequence~is~random$ versus $H_1:~The~sequence~is~not~random$
$n=48$, $n_N=22$, $n_S=26$ and $r=28$
Large sample case:
$µ_r=\frac{2n_1n_2}{n}+1=\frac{2\times22\times26}{48}+1=24.83$
$σ_r=\sqrt {\frac{2n_1n_2(2n_1n_2-n)}{n^2(n-1)}}=\sqrt {\frac{2\times22\times26(2\times22\times26-48)}{48^2(48-1)}}=3.40$
$z_0=\frac{r-µ_r}{σ_r}=\frac{28-24.83}{3.40}=0.93$
$z_{\frac{α}{2}}=z_{0.025}$
If the area of the standard normal curve to the right of $z_{0.025}$ is 0.025, then the area of the standard normal curve to the left of $z_{0.025}$ is $1−0.025=0.975$
According to Table V, the z-score which gives the closest value to 0.975 is 1.96.
Also, $-z_{\frac{α}{2}}=-1.96$
Since $-z_{\frac{α}{2}}\lt z_0\lt z_{\frac{α}{2}}$, we do not reject the null hypothesis.
