Hypothesis Testing: Null vs. Alternative Hypotheses

When we do a statistical test, there are two sides we can test for. The null hypothesis ($H_0$) states that there is no change; the mean given already is good enough. The alternative hypothesis states that there is a change. The alternative hypothesis ($H_a$) can state that the true mean, if it is not the mean already given, is either greater or smaller than the mean given, or one of the two. Here's an example which states the null and alternative hypotheses:

It is given that the mean IQ of the Googly Aliens is 276. However, a sample of 50 Googly Aliens gives a mean IQ of 190. We want to test if the mean IQ is actually lower than 276. The null hypothesis is given by $$H_0=276.$$ It basically says, nothing is wrong here, the mean IQ is actually 276. The alternative hypothesis is given by $$H_a<276,$$ which basically says, whoa, wait up here, the mean is actually less than 276. It is important to state the null and alternative hypotheses before performing a test of significance (aka a hypothesis test).

Statistical Tests (Significance Tests vs. Hypothesis Tests)

Statistical significance tests and hypothesis tests are often terms that are used interchangeably (often with the intention of actually referring to hypothesis tests).

Significance tests were developed first by Fisher, then later developed into hypothesis tests. Significance tests did not use language such as "reject the null" "fail to reject the null" but rather viewed the p-value as a continuous measure of evidence against the null hypothesis without actually creating a formalized decision making process.

It was not until later that Neyman & Pearson thought you could use the p-value as part of a formalized decision making process, introducing confidence/significance levels "alpha" $\alpha$ and a set of rules involving the p-value to either reject or fail to reject the null .

Hit "Continue" for explanations of both significance testing and hypothesis testing.

Significance testing (The Test To Guide Research) was developed by Fisher to try and quantify evidence against the null hypothesis ($H_0$) being true or false by examining something called a p-value.

Reason being, we believe our null hypothesis statement ($H_0$) may be true and hence we want to work out whether the statement $H_0$ is worth investing time making conclusions about it.

In order to ensure that we are confident in our statement $H_0$, we look at the opposite of this statement, the alternative hypothesis ($H_A$).

• If $H_A$ is quite likely (small p-value) then further study assuming $H_0$ to be true would seem to not make sense as the two hypothesis statements are mutually exclusive (cannot both occur at the same time). In other words, if $H_A$ is very likely (small p-value), $H_0$ must then be very unlikely we have more evidence to "reject the null hypothesis".

• However if $H_A$ is extremely unlikely (large p-value) then $H_0$ is probably extremely likely to occur and further study assuming $H_0$ to be true would make sense and hence we would have more evidence to "fail to reject the null hypothesis".

Very similarly in methodology, but with different use cases and an added "alpha" variable...

Hypothesis testing (The Test To Guide Decision Making) developed by Neyman and Pearson looks at evidence that supports ($H_0$) being true.

This is done by setting up both a null hypothesis and an alternative hypothesis, choosing an acceptable rate of allowable error "alpha" (usually 0.05), then either accepting or rejecting the null hypothesis based on whether the p-value is above or below alpha.

• if $p-value < \alpha$, reject the null hypothesis

• if $p-value > \alpha$, fail to reject the null hypothesis

We need to make a decision soon, and suspect that our null hypothesis $H_0$ is true and will greatly influence our decision. Thus we see how unlikely we are to be wrong by examining the likelihood of the alternative hypothesis ($H_A$), and if we are sufficiently unlikely to be wrong we can assume the null statement $H_0$ is true and follow through confidently with our final decision.

Null vs Alternative Hypothesis Statements

The hypothesis test is used to evaluate two mutually exclusive statements about a population to determine which statement is best supported by the sample data.

The alternative hypothesis ($H_A$) is the researcher's predicted difference from the population mean.

The null hypothesis ($H_0$) represents the opposite of the predicted difference from the population mean.

Below is an example of a two-tailed story problem being translated into hypothesis statements:

Two-Tailed Hypothesis Test Story Problem (Just Testing for a Difference):

A simple random sample of 10 people from a certain population has a mean age of 27. Can we conclude that the mean age of the population is not 30? The variance is known to be 20. Let alpha = 0.05

Because we are simply testing for a difference between the population mean (30) and sample mean (27), this is a two-tailed test. We will hence use the = and $\neq$ operators for the null and alternative hypothesis.

Step 1) State the null hypothesis ($H_0$) (what would happen if there is no difference between the population mean and sample mean)

$H_0: \mu = 30$ (population mean)

Step 2) State the alternative hypothesis (for one sided the operator is $\neq$)

$H_A: \mu \neq 30$ (population mean)

(Video) Null and Alternative Hypotheses

In this video by the University of Nottingham, you will learn the difference between null and alternative hypotheses.