Difference Between Null & Alternative Hypothesis (3 Key Ideas)

When using hypothesis testing in various fields (such as medicine or psychology), we often need to formulate both a null hypothesis and an alternative hypothesis.  It is important to know what they have in common and how they differ.

So, what is the difference between the null hypothesis and the alternative hypothesis?  The null hypothesis is a default assumption (that a parameter has a certain value, two populations are the same, or a treatment has no effect). The alternative hypothesis assumes otherwise (that a parameter does not equal a certain value, two populations are different, or a treatment has an effect).

Of course, the null hypothesis will often include an equality sign (one of the <=, =, or >= symbols).  The purpose of a hypothesis test is to reject (or fail to reject) the null hypothesis.

In this article, we’ll talk about the difference between the null hypothesis and the alternative hypothesis.  We’ll also look at some examples of each, along with what it means to reject the null hypothesis.

Let’s get started.

Difference Between Null & Alternative Hypothesis

The difference between the null hypothesis and the alternative hypothesis is that they make opposite claims.

The null hypothesis, H₀, is a default assumption about a quantity to be measured (such as the value of a parameter, the difference between two populations, or the effect of a treatment).

The alternative hypothesis, H_a, is the opposite of the null hypothesis, and it suggests that the default assumption is not correct.

For example, if a statistical test fails to reject the null hypothesis, then we can say with some level of confidence that:

• The assumed value of the parameter is correct.
• There is no difference between the two populations being compared.
• There is no effect from a particular treatment.

On the other hand, if a statistical test allows us to reject the null hypothesis in favor of the alternative hypothesis, then we can say with some level of confidence that:

• The assumed value of the parameter is not correct.
• There is a difference between the two populations being compared.
• There is an effect from a particular treatment.

Of course, before we can draw any conclusions, we need to take a few steps, including:

• Decide on a population to test (there may be two if we are comparing across populations)
• Decide on the population parameter to test (again, there may be two if there are two populations).
• Formulate (write) a null hypothesis, H₀.
• Formulate (write) an alternative hypothesis, H_a.
• Decide on a significance level.
• Calculate a test statistic.

Let’s talk about how to write a null hypothesis.

How To Write A Null Hypothesis

To write a null hypothesis, we need a parameter and a value to test.  The parameter is measuring some aspect of a population (or in some cases, the difference between two populations).

For example, the parameter could be:

• The population mean (average height of French men, average weight of cherry tomatoes).
• The difference between two populations (difference in average height between French men and Spanish men).
• The effect of a treatment (average change in blood cholesterol after a course of medicine).

If we suspect that the population mean has a certain value, then we can use that in our null hypothesis:

• The average height of French men is 70 inches.
• The difference in average height between French men and Spanish men is 0 inches.
• A treatment lowers blood cholesterol by 20 points.

To write a null hypothesis, we use the symbol H₀ (which stands for “H-nought”, “H-zero”, or “H-null”, the null hypothesis), followed by the hypothesis itself (usually that a parameter equals some value).

What Is An Example Of A Null Hypothesis?

Here are a few examples of null hypotheses.

Example 1: Average Height Of French Men

For the height of French men, our null hypothesis might be:

• H₀: The average height of a French man is 70 inches (h = 70).

Example 2: Difference In Average Height Between French Men & Spanish Men

For the difference in average height between French men and Spanish men, our null hypothesis might be:

• H₀: The mean difference in average height between French men and Spanish men is 0 inches (F – S = 0).

Example 3: Change In Blood Cholesterol After Treatment

For the cholesterol treatment, our null hypothesis might be:

• H₀: The mean change in blood cholesterol after treatment is 0 (D = 0).

In its simplest form, the null hypothesis can be written in the form:

• H₀: parameter = value

However, this assumes that the population and the parameter being measured are both clear.

A more formal way to write the null hypothesis is in the form:

• For the population A, the parameter P has a value of V (P = V).

This raises the question of whether the null hypothesis is always equal.

Is The Null Hypothesis Always Equal?

By convention, the null hypothesis H₀ is often a statement that involves equality, so that it includes one of the symbols <=, =, or >=.  However, this is not a requirement.

It is merely convenient that we formulate the null hypothesis by using an equality statement.

How To Write An Alternative Hypothesis

To write an alternative hypothesis, all we need to do is write the opposite of the null hypothesis:

• If the null hypothesis states that a parameter equals a certain value, then the alternative hypothesis would state that the parameter does not equal that value (we could specify greater than or less than).
• If the null hypothesis states that two populations show no difference in a parameter, then the alternative hypothesis would state that the two populations do show a difference in that parameter (again, we could specify which population has the greater value for the parameter).
• If the null hypothesis states that a treatment does not have an effect, then the alternative hypothesis would state that the treatment does have an effect (we can specify whether the treatment is positive or negative).

What Is An Example Of An Alternative Hypothesis?

Here are a few examples of alternative hypotheses.

Example 1: Average Height Of French Men

For the height of French men, our alternative hypothesis might be:

• H_a: The average height of a French man is less than 70 inches (h < 70).

Compare this to the null hypothesis that we formulated earlier:

• H₀: The average height of a French man is 70 inches (h = 70).

Example 2: Difference In Average Height Between French Men & Spanish Men

For the difference in average height between French men and Spanish men, our null hypothesis might be:

• H_a: The mean difference in average height between French men and Spanish men is more than 0 inches (F – S > 0).

Compare this to the null hypothesis that we formulated earlier:

• H₀: The mean difference in average height between French men and Spanish men is 0 inches (F – S = 0).

Example 3: Change In Blood Cholesterol After Treatment

For the cholesterol treatment, our null hypothesis might be:

• Ha: The mean change in blood cholesterol after treatment is less than 0 (D < 0).

Compare this to the null hypothesis that we formulated earlier:

• H₀: The mean change in blood cholesterol after treatment is 0 (D = 0).

When To Reject The Null Hypothesis

When our test statistic shows that the significance level is below our chosen threshold, we reject the null hypothesis.

For example, let’s say that after calculating a test statistic, we find that it is significant at the 2% level.

If our null hypothesis of H = 70 is correct, then there is only a 2% chance that we would have gotten the result we did.  If our threshold is 0.05 (95% significance), then we reject the null hypothesis in favor of the alternative hypothesis (that H > 70).

This would mean that we are fairly confident that the height of French men is in fact greater than 70 inches (rather than exactly 70 inches).

What Does It Mean To Reject The Null Hypothesis?

When we reject the null hypothesis, it means that the evidence against that hypothesis is too strong to ignore.  The definition of “too strong” depends on the tolerance level (significance level) that we choose.

• A 5% significance level is fairly strong.
• A 1% significance level is very strong.
• A 0.1% significance level is very difficult to ignore.

When To Accept The Null Hypothesis

We do not accept the null hypothesis.  Instead, we use the following language: “we fail to reject the null hypothesis.”

In this case, we are concluding that the evidence against the null hypothesis is not strong enough to reject it.

This doesn’t necessarily mean that the null hypothesis is true.  It just means that we are not confident enough to reject it and formulate another hypothesis.

If we gather more data and get a larger sample size, we may produce a test statistic that gives us a significant result.  Of course, we may also come to the same conclusion and fail to reject the null hypothesis.

Conclusion

Now you know the difference between the null hypothesis and the alternative hypothesis.  You also know how to formulate each one and what it means to reject the null hypothesis.

You can learn more about the difference between a parameter and a statistic in this article.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

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~Jonathon