Making conclusions in a z test for a proportion

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 $$c$$?

1 Answer

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)

• I don't understand what is meant by your first point. Also I don't see the words in point two in any of the choices. – Jinzu Dec 25 '18 at 0:16
• You found $p=0.025$. That would have been significant with $\alpha=0.05$ as in (c) and (d) but not with $\alpha=0.01$ as in (a) and (b). So that leaves (b) and (c) as possible conclusions. (c) has "conclude that the proportion has changed" which is essentially "reject the null hypothesis that the proportion has not changed" so looks reasonable, while (b) has "the proportion is still $10\%$" which is essentially "accept the null hypothesis of no change" so not reasonable in conventional hypothesis testing – Henry Dec 25 '18 at 8:48
• Is this because $.05$ is the standard for this type of testing? Because isn't it if the p value is higher than the significance level you accept the null hypothesis? – Jinzu Dec 25 '18 at 14:36