### Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (Critical Region Approach

### Formulating the Research Hypotheses

The first step of any statistical test is to formulate the research *hypotheses.*#\phantom{0}#

Hypothesis

In the context of statistical testing, a **hypothesis** is an expectation or prediction about a population characteristic.

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Every statistical test has two hypotheses: a *null hypothesis *and an *alternative hypothesis*.

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Null Hypothesis

The **null hypothesis** of a test, denoted #H_0#, represents a *zero-effect*. That is, any observed pattern in the sample data is purely due to *chance*.

Generally speaking, the null hypothesis is the hypothesis that you are trying to *disprove*.

The null hypothesis is the statement about the population that is being tested. The outcome of a hypothesis test always leads to one of two decisions:

- The null hypothesis is rejected.
- The null hypothesis is not rejected.

The null hypothesis should only be rejected if it is *highly unlikely *to be true given the observed sample data.

An important caveat is that *failing to reject* the null hypothesis does not mean the null hypothesis is accepted to be true. It simply means that there is not enough evidence to disprove it.

Examples of null hypotheses are:

- There is no treatment effect
- There is no difference between groups
- There is no relationship between variables

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Alternative Hypothesis

The **alternative hypothesis **of a test, denoted #H_a#, predicts that there *is *an effect. That is, any observed pattern in the sample data is not the result of random variation, but rather a reflection of the underlying population's characteristics.

Generally speaking, the alternative hypothesis is the hypothesis you suspect to be *true*.

The alternative hypothesis cannot be tested directly. It is accepted if and only if the null hypothesis is rejected.

The null hypothesis and the alternative hypothesis are *mutually exclusive*, meaning there cannot be any overlap in the predictions they make.

Examples of alternative hypotheses are:

- There is a treatment effect
- There is a difference between groups
- There is a relationship between variables

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Resemblance to a Criminal Trial

Hypothesis testing bears a resemblance to a criminal trial. In court, the burden of proof is on the prosecutor, meaning that a suspect is assumed to be innocent until proven guilty.

Likewise, in hypothesis testing, the null hypothesis is assumed to be true unless very strong evidence is presented that suggests otherwise.

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2. Research Hypotheses

The null hypothesis should state the Summer Course has no effect on the students' grades. If this is true, then the mean grade of the students who take the Summer Course will be equal to the mean grade of the students who have not taken the Summer Course.

The alternative hypothesis simply covers all other outcomes.

\[\begin{array}{c}

H_0: \mu_{\text{after summer course}}=6.5\\

H_a: \mu_{\text{after summer course}}\neq 6.5\\

\end{array}\]

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