Reject null hypothesis or not?

Question

Tony conducts a hypothesis test on whether the proportion of all students in his highschool who bike to school (denoted as $p$) equals $30\%$. Specifically, Tony has $H_0:p=0.3$ versus $H_A:p≠0.3$. He obtains a P-value of $0.01$. On the other hand, John would like to test if there is sufficient evidence to support that $p$ is greater than $0.3$ at the $10\%$ significance level. Based on Tony's result, will the null hypothesis of John's test be rejected?

Here is what I have done.

Tony: $H_0:p=0.3$ , $H_A:p≠0.3$ with P-value of $0.01$

John: $H_0:p=0.3$ , $H_A:p>0.3$ with $\alpha = 0.1$

From Tony's result, p-value = $0.01 $, which is less then $\alpha = 0.1$. Hence the null hypothesis of John's test should be rejected.

Do I have the right procedures?

Answer 1

There is not enough information to decide whether it should be rejected.

For a two-tail p-value(0.01 in this case), half of that p-value is located at each tail(ie-.005 at each tail).

For a one-tail p-value, you would only consider one tail, resulting in a p-value of .005 or 0.995, depending on whether the observed proportion was greater than or less than 0.3.

Furthermore, Johns null hypothesis should be less than or equal, not just equal. This is because you never prove an alternative hypothesis correct, you prove the null hypothesis incorrect.

Answer 2

Both results are compatible. There are two different results and conclusions for two different $H_0$. (what $H_0$ is more reasonable is another question).

If the sample size was enough, for example of $n=500$ students, and the proportion was only $p=0.01$, for example, certainly Tony will be sure (without any test) that true proportion is not equal to $0.3$ --> $H_0$ rejected. And John, also a clever man, will note that without any test is also clear that true proportion must be near 0.01 and therefore cannot be greater in any case more than $0.3$ --> $H_0$ accepted. What is the problem here?

And what about if the proportion of the sample was $p=0.33$? Then the brain calculation of Tony is not reliable, but make a correct test in R with a bigger sample: according to his $H_0$ and the results is a p-value > 0.05$

prop.test(200,600,c(.3), alternative ="two.sided")

While John make this according to their $H_0$:

 prop.test(200,600,c(.3), alternative ="greater")

And the p-value was > 0.05.

Tony result cancel Jhon results, or at the contrary in this case? No. The apparent difference is not. The mistake here is consider that Tony have tested only if p is greater than 0.3. Also have tested if p is less than 0.3. For Tony there are two sources of error ($\alpha$ are in two tails) and this is too much to reject safely $H_0$, while Jhon's hypothesis have only a possible tail of error ($\alpha/2$) since he is comparing only if p is greater than 0.3. Therefore Jhon can reject his $H_0$ and Tony cannot, because is not the same hypothesis.

Moreover, it is worth to remember that reject $H_0$ means that you have demonstrated (with little margin of error) that $H_1$ is true. However, if accept your $H_0$ means nothing. Only that you have not demonstrated that $H_1$ is true. But this does not mean that $H_1$ is false, (may be with a bigger sample...) nor that $H_0$ true, and therefore, in any way than another $H_0$ is false or true.

Answer 3

Almost but not quite. Tony is testing a two tailed hypothesis; John is not. You need to explicitly consider what impact that change has.

Answer 4

I'll just introduce the following example:

Imagine that Tony rejects the H0, because the proportion is significantly below 0.3.

There is then no way that John would be able to reject the null hypothesis, because he is expecting an effect above 0.3.

In the other case, the case in which the proportion is significantly above 0.3, we can be sure that John would reject the null as well, since it is in his direction (and he has an even easier time rejecting the null since he is allowed to divide the alpha by doing a one tailed test).