1. ## Hypothesis Testing

I'm not sure how to work out this question:

The Australian Medical Association believed that the Health Minister's recent statement claiming that at least 80% of doctors supported the reforms to Medicare was incorrect. The Association's President suggested the best way to test this was to survey 200 members, selected through a random sample, on the issue. She indicated that the Association would be prepared to accept a Type I error probability of 0.02.

1. State the direction of the alternative hypothesis for the test.

2. Using the tables, state the absolute value of the test statistic

and determine the lower boundary 3._____, and the upper boundary 4______.
of the region of non-rejection in terms of the sample proportion of respondents (as %) in favour of the reforms. If there is no (theoretical) lower bound, type lt in box 3, and if there is no (theoretical) upper bound, type gt in box 4. State the numerical value(s) correct to one decimal place.

5. If 137 of the survey participants indicated support for the reforms, is the null hypothesis rejected for this test? Type yes or no.

6. If the null hypothesis is rejected, can the Association claim that the Health Minister's assertion is incorrect at the 2% level of significance on the basis of this test?

Thank-you very much for your help!!

2. Not everyone does these the same way.
I look at hypothesis testing as a proof by contradiction.
I assume the null hypothesis to be correct and see if the data tells me it's wrong.
Hence I want to prove the alternative hypothesis.
In this case we want to prove that .8 is too high, hence

$H_0=.8" alt="H_0=.8" /> vs. $H_a<.8" alt="H_a<.8" />

This absolute vale, seems to hint at a two-sided test, whihc I don;t agree with.

My test stat would be

$Z^*={\hat p-p_0\over \sqrt{p_0q_0/n}}$

which is close to a normal since n=200.

Here $\hat p=137/200$ and $p_0=.8$

3. This would be a one tail test, and would be left tailed. A two tailed test would be if the statement was "The proportion of so-and-so is EXPLICITLY equal to some number"; thus your null hypothesis would fail if it were greater than, or less than some critical values. For example, if some machine process had to correct to some specific number, no more no less - a test on this process would be two tailed since it could be rejected if it feel within the greater than or less than region (if that makes any sense). Here we are saying the population proportion is at LEAST some value, which means the null hypothesis only fails if if we get values that are less than that value. Two tails would allow for the null hypothesis to fail if we got a value greater than 0.8, which doesn't make sense in the context of the problem.

They say absolute value, because your critical value will have a z-score to the right of the population proportion and thus would be negative.

I agree with the rest, except my null hypothesis would be $h_0: p \ge 0.8$

4. The two procedures...

$H_0=.8" alt="H_0=.8" /> vs. $H_a<.8" alt="H_a<.8" />

and

$H_0\ge .8" alt="H_0\ge .8" /> vs. $H_a<.8" alt="H_a<.8" />

are exactly the same.
In the second one, $\alpha$ is the largest (sup) probability
of committing a type one error over the region $p\ge .8$
which occurs at the boundary of $p= .8$
I stick with simple hypotheses to make it easier on the students.

5. The only reason I suggest people use the less than or greater than signs in their null hypothesis, is because they may look at it and think "two tailed test, since the opposite of equals is 'not equals.' " As you said, it will all lead to the same answer so, no harm no foul.