A poll from a previous year showed that $10\%$ of smartphone owners relied on their data plan as their primary form of internet access. Researchers were curious if that had changed, so they tested $H_0: p=10\%$ versus $H_a:p\neq10\%$ where $p$ is the proportion of smartphone owners who rely on their data plan as their primary form of internet access. They surveyed a random sample of $500$ smartphone owners and found that $13\%$ of them relied on their data plan.
The test statistic for these results was $z\approx 2.236$, and the corresponding P-value was approximately $0.025$.
Assuming the conditions for inference were met, which of these is an appropriate conclusion?
$a)$ At the $\alpha=0.01$ significance level, they should conclude that the proportion has changed from $10\%$.
$b)$ At the $\alpha=0.01$ significance level, they should conclude that the proportion is still $10\%$.
$c)$ At the $\alpha=0.05$ significance level, they should conclude that the proportion has changed from $10\%$.
$d)$ At the $\alpha=0.05$ significance level, they should conclude that the proportion is still $10\%$.
The correct answer is $c$ but why could it not have been $b$? Why is it $c$?
Two issues:
(c) has $0.05$ while (b) has $0.01$
Your choices are conventionally "reject the null hypothesis" or "do not reject the null hypothesis", but not "accept the null hypothesis" which is close to the wording of (b) and (d)