Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population, or from a data-generating process. The word “population” will be used for both of these cases in the following descriptions.

### KEY TAKEAWAYS

- Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
- The test provides evidence concerning the plausibility of the hypothesis, given the data.
- Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.

## How Hypothesis Testing Works

In hypothesis testing, an analyst tests a statistical sample, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis; e.g., the population mean return is not equal to zero. Thus, they are mutually exclusive, and only one can be true. However, one of the two hypotheses will always be true.

### Four Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

- The first step is for the analyst to state the two hypotheses so that only one can be right.
- The next step is to formulate an analysis plan, which outlines how the data will be evaluated.
- The third step is to carry out the plan and physically analyze the sample data.
- The fourth and final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

## Real-World Example of Hypothesis Testing

If, for example, a person wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be yes, and the alternative hypothesis would be no (it does not land on heads). Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as “Ha” and be identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is “accepted,” the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is “explainable by chance alone.”

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1. Hypothesis testing: how to form hypotheses (null and alternative); what is the meaning of reject the null or fail to reject the null; how to compare the p-value to the significant level (suchlike alpha = 0.05), and what a smaller p-value means.

2. How to interpret the one-sample t-test results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value; what are the steps for the one-sample t test; what a normal distribution looks like.

3. How to interpret the one-way ANOVA results: what are Ho and Ha; the standard for determining statistical significance, i.e., F statistic and p-value; what an F distribution looks like.

4. How to interpret the simple linear regression results: what are Ho and Ha; the standard for determining statistical significance, i.e., t statistic and p-value of the slope; what is the slope and what it means; what is the R-square (not R, it is R-square!) and what it means; what are independent variables and dependent variable, and what their relationships are; how would you plot the relationship between a dependent variable and an independent variable; from a given independent variable, how would you predict the value of a dependent variable.

5. How to interpret the multiple regression results: how to interpret the slope of an independent variable (i.e., the impact of this independent variable, holding other independent variables constance).