Answer
$z_0\gt -z_α$: null hypothesis is not rejected.
There is not enough evidence to conclude that less than 50% of gamers are women.
Work Step by Step
$H_0:~p=0.5$ versus $H_1:~p\lt0.5$
$n=100$ (large sample)
$number~of~plus~signs=38$ (women)
$number~of~minus~signs=100-38=62$ (men)
Left-tailed test:
$k=number~of~plus~signs=38$
$z_0=\frac{(k+0.5)-\frac{n}{2}}{\frac{\sqrt n}{2}}=\frac{(38+0.5)-\frac{100}{2}}{\frac{\sqrt {100}}{2}}=-2.3$
$z_α=z_{0.01}$
If the area of the standard normal curve to the right of $z_{0.01}$ is 0.01, then the area of the standard normal curve to the left of $z_{0.01}$ is $1−0.01=0.99$
According to Table V, the z-score which gives the closest value to 0.99 is 2.33.
So, $-z_α=-2.33$
Since $z_0\gt -z_α$, we do not reject the null hypothesis.
