The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some cause that we should be looking for? We need to have a way to differentiate between events that easily occur by chance and those that are highly unlikely to occur randomly. Such a method should be streamlined and well defined so that others can replicate our statistical experiments.
There are a few different methods used to conduct hypothesis tests. One of these methods is known as the traditional method, and another involves what is known as a p- value. The steps of these two most common methods are identical up to a point, then diverge slightly. Both the traditional method for hypothesis testing and the p-value method are outlined below.
The Traditional Method
The traditional method is as follows:
- Begin by stating the claim or hypothesis that is being tested. Also form a statement for the case that the hypothesis is false.
- Express both of the statements from the first step in mathematical symbols. These statements will use symbols such as inequalities and equals signs.
- Identify which of the two symbolic statements does not have equality in it. This could simply be a "not equals" sign, but could also be an "is less than" sign ( < ) or an "is greater than" sign ( > ). The statement containing inequality is called the alternative hypothesis, and is denoted H1 or Ha.
- The statement from the first step that makes the statement that a parameter equals a particular value is called the null hypothesis, denoted H0.
- Choose which significance level that we want. A significance level is typically denoted by the Greek letter alpha. Here we should consider Type I errors. A Type I error occurs when we reject a null hypothesis that is actually true. If we are very concerned about this possibility occurring, then our value for alpha should be small. There is a bit of a trade off here. The smaller the alpha, the most costly the experiment. The values 0.05 and 0.01 are common values used for alpha, but any positive number between 0 and 0.50 could be used for a significance level.
- Determine which statistic and distribution we should use. The type of distribution is dictated by features of the data. Common distributions include: z score, t score and chi-squared.
- Find the test statistic and critical value for this statistic. Here we will have to consider if we are conducting a two tailed test (typically when the alternative hypothesis contains a “is not equal to” symbol, or a one tailed test (typically used when an inequality is involved in the statement of the alternative hypothesis).
- From the type of distribution, confidence level, critical value and test statistic we sketch a graph.
- If the test statistic is in our critical region, then we must reject the null hypothesis. The alternative hypothesis stands. If the test statistic is not in our critical region, then we fail to reject the null hypothesis. This does not prove that the null hypothesis is true, but gives a way to quantify how likely it is to be true.
- We now state the results of the hypothesis test in such a way that the original claim is addressed.
The p-Value Method
The p-value method is nearly identical to the traditional method. The first six steps are the same. For step seven we find the test statistic and p-value. We then reject the null hypothesis if p-value is less than or equal to alpha. We fail to reject the null hypothesis if the p-value is greater than alpha. We then wrap up the test as before, by clearly stating the results.