#### Null Hypothesis

Now it is time to talk a bit about the null hypothesis. We said that the main idea of hypothesis testing is, simply put, a way of validating or evaluating the likelihood of a certain statements about a population parameter (typically \(\mu\) or \(\sigma\), etc.) Two possible statements about a population parameter are the null and alternative hypothesis.

The null hypothesis is usually written as \(H_0\) and alternative hypothesis is written as \(H_a\). In a few minutes it will be clear the reason for the names.

**Example:** Suppose that it is known that in the past, teachers made and average of $800 per week. You are interested in finding out whether or not that figure had changed or not. A sample of \(n=35\) teachers was randomly selected, and the results show that the sample mean is \(\overline{X} = 825\) and the sample standard deviation is \(s = 69\). What is the null and alternative hypothesis?

Here we apply a little rule. The null hypothesis corresponds to the statement that includes the condition of *no change* or *no difference*. If we carefully look at the situation above, there are two possibilities: Either the salaries have changed, or they haven’t changed. So, what is the null hypothesis? The one that includes the condition of no change. In this case, the null hypothesis is that claim that the salaries haven’t changed.

And now, what is the alternative hypothesis? This is even easier: The alternative hypothesis is the complement to the null hypothesis (or in other words, it states the **the contrary** to the null hypothesis). Summarizing, in the previous example we have that the null and alternative hypothesis are:

This kind of test is called *two-tailed*.

**Definition:** The null hypothesis \(H_0\) corresponds to the statement that includes the condition of no change, and the alternative hypothesis \(H_a\) corresponds to the opposite of the null hypothesis. In other words, the alternative hypothesis must be true when the null hypothesis is false (Mathematically, this means that the sets are disjoint, or, the null and alternative hypothesis cannot **overlap**.)

Now, there are several different situations, where we can find different types of null hypothesis.

**Example:** Imagine that you want to find out whether or not dogs live more than 15 years in average. You collect a sample of \(n=20\) records showing the age that the dogs lived, and the results show that the sample mean is \(\overline{X} = 16\) and the sample standard deviation is \(s = 4\). What is the null and alternative hypothesis?

Here, the trick is always the same: What is the statement that includes the condition of no change? Let’s examine the problem. The research hypothesis is that dogs live **more** than 15 years. This statement doesn’t include the no change condition. What’s the other option? Dogs live 15 years or less. And this last claim **does include** the no change condition. Hence, we have that

This kind of test is called *right-tailed*.

**Example:** Finally, assume that you are interested in testing whether or not your deodorant last less than 10 days in average. You record a random sample of 10 bars, and the mean is \(\overline{X}= 8\) and the sample standard deviation is \(s = 2\). What is the null and alternative hypothesis?

Using the same as before, we have that

\[H_0: \mu \geq 10 \] \[H_a: \mu < 10\]This kind of test is called *left-tailed*.

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