# Figuring out Which is Null and which is Alternative

After doing homework problems for stats, I am having a tough time for understanding which one is null hypothesis and which one is alternative hypothesis. Is there an intuitive explanation so that when I take my final exam, I will know which one is alternative or which is null. For example, in the picture attached, how did he figure out the hypothesis tests, because my null hypothesis was that p1 >= p2 and my alternative was that p1< p2.

In a study of patients on sodium-restricted diets, 55 patients with hypertension were studied. Among these, 24 were on sodium-restricted diets. Of 149 patients without hypertension, 36 were on sodium-restricted diets. We would like to know if we can conclude that, in the sampled population, the proportion of patients on sodium-restricted diets is higher among patients with hypertension than among patients without hypertension.

(1) Data

Patients with hypertension: $$\,\quad n_1 = 55,\,\:\quad x_1 = 24, \quad \hat{p}_1 = .4364\\$$ Patients without hypertension: $$n_2 = 149,\quad x_2 = 36, \quad \hat{p}_2 = .2416$$

$$\hspace{4cm}\alpha=\:\: .05$$

(2) Assumptions

• independent random samples from the populations

(3) Hypotheses

$$\begin{eqnarray} H_0\,:\: p_1 &\leq& p_2\\ H_1\,:\: p_1 &>& p_2 \end{eqnarray}$$

• $p_1>p_2$ is not a suitable null, because the limit of the parameter space under the null is not in the null space. (i.e. the equality case, $p_1=p_2$, which will be the case under which you compute your p-values, isn't in your null hypothesis). Please note that there was an equality in the quoted material, you just dropped it in your discussion. I have taken the liberty of retyping your image, as the text in the image was much too small to read (this also makes it at least somewhat accessible to vision-impaired people using screenreaders). Dec 9 '18 at 1:56
• Im still a bit confused how do you know that p1<= p2 and how do you determine what null hypothesis are when being asked Dec 9 '18 at 2:52
• Yes, but that's the province of an answer to the question; I was trying to get you to fix your question before people attempted to answer. Please fix your question Dec 9 '18 at 6:34

The null hypothesis is almost always "nothing is going on". E.g. "There is no difference between these means" (t-test, ANOVA) "The two variables aren't related" (regression) "All the types of subject survive just as long" (survival analysis) etc.

This is not always true. It is possible to write a null that posits some relationship. But in the vast majority of cases, the null will be as above.

To the extent that I encounter statistical tests I am usually able to interpret their underlying logic as having the form: I want to know $$A$$ hence I must try to statistically reject $$notA$$, where $$A$$ is some statement expressing my knowledge interest. Since it is the null-hypothesis that risks being statistically rejected my $$H_0$$ must be $$notA$$.

Therefore I always look after the knowledge interest: What it is they/I want to know? In your example this is revealed by the terms "We would like to know". The knowledge of interest should then be defined in some declarative statement, hence capable of being true or false. Since they want to know whether

"the proportion of patients on sodium-restricted diets is higher among patients with hypertension than among patients without hypertension"

this must be the alternative hypothesis $$A$$. The null hypothesis is then the negation or the opposite of this statement. Because, then if they can reject the null, they have succeeded in providing some empirical basis for believing the alternative hypothesis. They strengthen the believe in $$A=p_1>p_2$$ through the statistical rejection of the opposite $$H_0=p_1 \leq p_2$$.

Admittedly this is a very non-technical answer, but is has served me well for what it is worth.