8.4 – The Null and Alternative Hypotheses

As discussed earlier, a hypothesis or prediction is a critical component of the scientific method. By hypothesizing what we think will happen in a research study we are then able to test that prediction in our study to see if the hypothesis is supported by the data or not. This makes it harder for us to believe things just because we want to believe them. Instead, we need to see if our beliefs actually pass the test. Additionally, making a clear hypothesis before we run the test helps keep us from simply changing our hypothesis after the test to fit the results. The inclusion of the hypothesis in the scientific method and NHST is critical.

However, in null hypothesis significance testing, there are actually two hypotheses involved:

• Alternative Hypothesis (H₁ or H_a) – The hypothesis that the researchers want to explore. It predicts the relationship between at least two variables.

• Null Hypothesis (H₀) – The negation of the hypothesis that the researchers want to explore. It predicts that the relationship specified in the alternative hypothesis is not true.

Why are there two hypotheses? Because in the scientific method, researchers are expected to try to disprove their hypothesis (refer back to section 8.1 for the reasoning behind the focus on disproof), and that is why researchers are expected to create the null hypothesis. But, because the null hypothesis is the negation of the researcher’s hypothesis, the researchers need to specify what they are predicting will happen in their study, which is the alternative hypothesis. In other words, researchers would typically create their alternative hypothesis, and then use that to create the null hypothesis. And then, in the end, their research study (as we will see) will then actually only test the null hypothesis.

This fact also can help explain why the research hypothesis is referred to as an “alternative” hypothesis, even though it seems to be the primary prediction of the researchers. Instead, it is an “alternative” hypothesis because it is not the hypothesis that will actually be tested in the research process.

You’ll, hopefully, note that the alternative hypothesis (H₁) and the null hypothesis (H₀) map well to our two possible conclusions from our example research study on the treatment for anxiety. The alternative hypothesis is essentially what the researchers typically are expecting to happen — they think the treatment will reduce anxiety levels. The null hypothesis, on the other hand, is that the treatment will not reduce anxiety levels:

• H₁: Treatment reduces anxiety.
• H₀: Treatment does not reduce anxiety (any reductions are due to chance).

Then, the goal of using inferential statistics is to try to figure out which of these two hypotheses (or possible explanations) is more likely to be the best explanation for the results of our research study.

Types of Relationships in Research Hypotheses

You may note that both the alternative and null hypotheses involve predictions about relationships between variables. However, there are a number of different types of relationships that can be predicted, and the type of relationship depends on the research question (which also determines the methodology used in the research study as well as the appropriate inferential statistic).

Throughout the rest of the textbook, we will explore a number of different inferential statistics, and you will see that each test statistic was designed for different research questions that predict specific types of relationships between variables. Here are some examples of the different types of relationships that we will learn to test throughout this textbook:

Formulating the Alternative Hypothesis in Sentence Form

In order to create our hypotheses (alternative and null), it is often best to start with the alternative hypothesis, which is the one that the researchers would predict (note: for most questions in homework and exams, it will usually be described in the question). So let’s use our example regarding our new treatment for anxiety, where the researchers believe that this new treatment will reduce anxiety. This is an example of a hypothesis that is predicting the effect of one variable on another variable, so we would be using the format of “Variable A will have some kind of effect (increases/decreases) on Variable B.” Thus, the alternative hypothesis would be:

• Alternative Hypothesis: “The new treatment will reduce anxiety”.

By stating their alternative hypothesis before running their research study, the researchers are being direct and transparent about what they hope will happen. This makes it harder, then, to try to change their prediction if they were to run their study and find that the results did not show a reduction in anxiety.

Directional (One-Tailed) vs. Non-Directional (Two-Tailed) Hypothesis

When determining the alternative hypothesis, researchers have the option of choosing between a directional hypothesis or non-directional hypothesis. These are also often referred to as a one-tailed hypothesis (directional) or a two-tailed hypothesis (non-directional).

Technically, the alternative hypothesis above is a one-tailed or directional hypothesis because it specifies the direction of effect that the new treatment will have on anxiety – a reduction (or in the negative direction). If, alternatively, the researchers wanted to predict a two-tailed hypothesis, they would state it as:

• Alternative Hypothesis: “The new treatment will affect anxiety.”

By stating that the treatment will only “affect” anxiety, they are not specifying the direction of the effect and, thus, this is an example of a non-directional or two-tailed hypothesis. In this case, the researchers are predicting that the new treatment will either reduce or increase anxiety.

Formulating the Null Hypothesis in Sentence Form

Once researchers have stated their alternative hypothesis, they can then create the null hypothesis. It is important to remember that the null hypothesis is the negation (“not”) of the alternative hypothesis. It is not the “opposite” of the alternative hypothesis because it must include any possible outcome that negates the alternative hypothesis. Let’s start with our original alternative hypothesis:

• Alternative Hypothesis: “The new treatment will reduce anxiety”.

In this anxiety treatment example, there are technically two outcomes that negate this hypothesis that the treatment reduces anxiety:

1. The treatment increases anxiety (this is the “opposite”).
2. The treatment has no impact on anxiety (in other words, anxiety didn’t change at all).

Neither of these outcomes supports the alternative hypothesis that the treatment reduces anxiety. The null hypothesis, ultimately, includes any result that would support the disproof of the alternative hypothesis.

Thus, the null hypothesis would be:

• Null Hypothesis: “The new treatment will not reduce anxiety.”

Notice, that it is basically the same sentence as the alternative hypothesis, but with the word “not” added before the verb describing the effect (in this case, “reduce”).

This negation idea is important because you will see that we are actually going to focus on the null hypothesis when we follow our hypothesis testing system. In other words, as the scientific method requires, researchers attempt to disprove their hypotheses rather than prove them. As we saw with our discussion of confirmation bias, the best way to get closer to the “truth” is to focus on attempting to disprove our hypotheses, and, ultimately, the null hypothesis predicts the disproof of the alternative hypothesis.

Now, let’s write the null hypothesis for our two-tailed alternative hypothesis. Again, we start with the alternative hypothesis:

• Alternative Hypothesis: “The new treatment will affect anxiety.”

Again, the null hypothesis is the negation of this hypothesis. It’s often helpful to simply think of adding the word “not” before the verb in the hypothesis. Thus, we would end up with:

• Null Hypothesis: “The new treatment will not affect anxiety.”

In this case, the disproof of our alternative hypothesis is a result that shows that anxiety is unchanged by the new treatment.

Template for Hypotheses in Sentence Form

You may pick up that our hypotheses follow a similar pattern in their structure. Typically, the sentences include both of the variables that are predicted to be related to each other, and then the type of relationship is described as a directional or non-directional relationship using a verb. Thus, you have a sentence like this:

“[Variable A] [Type of Relationship] [Variable B]”

For now, these are our options for writing alternative hypothesis sentences because we will be working with research studies that are focused on examining the causal impact of one variable on another variable. However, in the future, our research studies will involve different types of relationships. Thus, the hypotheses make take slightly different forms:

Formulating Alternative Hypothesis in Symbol Form

Now that you have been shown how to write the alternative hypothesis in sentence form, we are now going to explore how to write the alternative hypothesis in symbol form. While the sentence form is helpful to understand what the researchers are predicting to happen in their study, we can also use the hypotheses to predict numerical results, which is necessary for calculating our inferential statistics. That’s where writing hypotheses in symbol form is helpful.

When we write the alternative hypothesis using symbols we will typically be making a prediction about a population parameter. For now, we will be hypothesizing about the parameter: population mean (μ).

We focus on population parameters in our hypotheses because our research hypotheses are technically making predictions about populations of individuals and not just the participants in a given research study. So when we predict that “the new treatment will reduce anxiety,” we are actually making a prediction about a hypothetical situation where we give everyone who has anxiety this new treatment, and thus we predict what will happen to their anxiety.

Let’s assume that we have an anxiety measure that we will use to measure the anxiety levels of our participants called the Very Useful Anxiety Scale (VUAS), and we know that on this measure anxious people have an average anxiety score of μ = 50. As a result, if we then hypothetically had every anxious person receive the new treatment, we can now make a prediction about that population mean (μ) based upon our original alternative hypothesis in sentence form:

• Alternative Hypothesis: “The new treatment will reduce anxiety”.

If this hypothesis is correct, then anxious people receiving the new treatment should have a lower anxiety score on the Very Useful Anxiety Scale than those who have not received the new treatment. Because we know that those who have not received the treatment have a mean of μ = 50, we can now state the alternative hypothesis using symbols:

• H₁: μ_{new treatment} < 50

What this is stating is that a hypothesized population of people who received the new treatment would have an average anxiety score of less than 50 on the VUAS. In other words, the new treatment is predicted to reduce anxiety.

Also note that instead of writing out the words “alternative hypothesis,” we instead use the symbol for the alternative hypothesis, H₁:

Template for Hypotheses in Symbol Form

As with stating hypotheses in sentence form, hypotheses in symbol form also follow a pattern. First, they will include a prediction about a particular population mean (μ), so they will include the symbol μ, and then in the subscript of that population mean it will specify the condition under which we are hypothesizing about that population mean, which will be one of the variables (in this case, Variable A). Usually, this will be the independent variable because we are looking at whether the independent variable has an impact on the dependent variable. The population mean (μ) in the hypothesis depicts what is being measured (in this case, Variable B). Usually, this will be the dependent variable because it is the variable that is measured. We are then going to predict, using a symbol of inequality (<, >, or ≠), how different this variable will be from its status quo (the population mean of the dependent variable not under the condition of the independent variable). Thus, the template takes this general form:

“μ[Variable A] [> or < or ≠] [Untreated μ for Variable B]”

Let’s return to the alternative hypothesis for our study of the impact of the new treatment (Variable A / Independent Variable) on anxiety (Variable B / Dependent Variable):

• H₁: μ_{new treatment} < 50

This hypothesis can then be interpreted as follows. This alternative hypothesis (H₁) predicts that the average anxiety score (μ) of people receiving the treatment (_{new treatment}) will be less than (<) the average anxiety score of people who don’t receive the treatment (which we happen to know is 50).

Now, we can write the null hypothesis in symbol form:

• H₀: μ_{new treatment} ≥ 50

Hopefully, you can see that the null hypothesis is the negation of the alternative hypothesis. Because the alternative hypothesis predicts anxiety scores less than 50, any scores not less than 50 would negate (or disprove) that hypothesis. Thus, it includes scores greater than (>) 50 as well as scores equal to (=) 50.