# Hypothesis Testing: How to Determine the Null and Alternative Hypotheses?

One thing that can be tricky when attempting to solve a hypothesis testing problem is to establish precisely what the null and alternative hypotheses are. Typically, such information can be easily inferred from the context of the problem, but you need to know what to look for in order to get it right.

### HOW TO GET STARTED

First thing to keep in mind is the precise specification of the null and alternative hypotheses can be inferred from the wording on the actual problem. Somewhere in the setting of the problem you will find where the hypotheses are stated.

Second, you need to keep in mind that the null and alternative hypotheses DO NOT OVERLAP. This implies that for the most part you can tell the null hypothesis if you know the alternative hypothesis, and vice versa, with some exceptions as we will see in the next paragraph.

Third, when reading the setting of an hypothesis testing problem, we need to identify any claim made about a population parameter, and express it in mathematical terms, such as \(\mu =2.3\), \(\mu \le 3\), \(\sigma >3.5\), etc. This is VERY IMPORTANT, because once we have expressed the claim(s) provided mathematically, we need to take note of which mathematical sign is used (\(\le\), \(\ge\), = , < or >).

The fourth point to keep in mind is the hypothesis of no effect, and it must contain the "=" sign, which means that the sign in the null hypothesis can be "\(\le\)", "=" or "\(\ge\)". And since the null hypothesis and alternative hypothesis cannot overlap, the only options for the sign of the alternative hypothesis are ">" or "<".

The above information should in fact be sufficient to determine the null and alternative hypothesis with ease.

### SOME PRACTICAL EXAMPLES

For example, say that we are examining a hypothesis testing question from our stats homework, and scanning the problem we read something like "and the investigator wants to prove if the average mileage for the new model is greater than 18 mpg". Such statement is a claim about the population mean mileage of the new car model, which we call \(\mu\).

The claim that the investigator is making is that "\(\mu >18\)". Since the mathematical expression of the claim does not contain "=", then the claim must be the alternative hypothesis. So then in this case we have the alternative hypothesis is Ha: \(\mu >18\). What is the null hypothesis then? Well, we know that the null and alternative hypotheses do not overlap, so we can say that the null hypothesis is the COMPLEMENT to what is expressed in the alternative hypothesis, so then in this case the null hypothesis is Ho: \(\mu \le 18\).

Therefore, summarizing, in this case the null and alternative hypotheses would be:

\[\begin{align} {{H}_{0}}:\mu \le 18 \\ {{H}_{A}}:\mu >18 \\ \end{align}\]

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Another example
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: Assume that the setting of the problem reads something like "a sample was collected to assess if the IQ of Stats professors is the same as national average IQ of 102". In that case, there is a claim about the population IQ of all Stats professor, which we shall call \(\mu\). The claim made is \(\mu =102\), and since this statement contains the sign "=", then this MUST be the null hypothesis. Hence, in this example we have that Ho: \(\mu =102\).

What is the alternative hypothesis then? Since the null and alternative hypotheses do not overlap, the alternative hypotheses is the complement to the null hypothesis, so in this case the alternative hypothesis would be $\mu \ne 102$.

Therefore, summarizing, in this case the null and alternative hypotheses would be:

\[\begin{align} {{H}_{0}}:\mu =102 \\ & {{H}_{A}}:\mu \ne 102 \\ \end{align}\]

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Another example: Things are not always that easy.
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Sometimes, things get a bit more complicated (but only a bit, I promise) when it comes down determining the null and alternative hypothesis from the setting of a question. Indeed, sometimes, there are actually two claims about a population parameter. For example, you start reading a question and you find the following: "it has been claimed that the population mean GPA for some state college is 3.94".

So you think, ok, the parameter is the population mean GPA for the state college, which we call \(\mu\), so then this statement is saying that \(\mu =3.94\), and since this mathematical statement contains the sign "=", then this must be the null hypothesis Ho. So we know for a fact that Ho: \(\mu =3.94\). Then you say, I can say that obviously the alternative hypothesis is Ha: \(\mu \ne 3.94\), right? Not so fast! If NOTHING else is claimed about \(\mu\) in the setting of the problem, then you can go and say that Ha: \(\mu \ne 3.94\).

BUT, sometimes another claimed is made. Indeed, suppose that in this case, you take a close look and you reread the problem, and it says " it has been claimed that the population mean GPA for some state college is 3.94, and a random sample has been collected to test the claim of the dean of the college, who claims that the mean GPA is less than that". Aha! In this case there is ANOTHER claim saying \(\mu <3.94\). And since this claim does NOT contain the sign "=", the it must be the alternative hypothesis. So then in this case, we get that Ha: \(\mu <3.94\) and not Ha: \(\mu \ne 3.94\).

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Should you be worried about seeing more than two claims in a problem involving hypothesis testing?
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The answer is NO. More than two claims will either lead to redundant or contradictory claims, for which reason you will likely not find such situation (unless the problem is wrongly posed, which is always a possibility). So then, when facing a problem, you will find one claim about a population parameter which will determine the null or alternative hypothesis, and you can deduce the other by using getting the complement of the given claim. OR, you will find two claims that do not overlap, which will define the null and alternative hypotheses.

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