Introduction to Hypothesis Testing
Suppose you want to know whether a coin is fair. How would you find out? You'd probably toss the coin a bunch of times and look to see whether heads shows up about 50% of the time. How far from 50% would the percentage of heads need to be to convince you that the coint is not fair? What if you only got 40% heads? Would it matter whether that 40% was out of 10 tosses or out of 1000 tosses? Why?
Questions like those above are at the heart of statistical methods called 'hypothesis tests'. A hypothesis test is used to determine
whether a parameter of interest is equal to some specified value.
In the coin example, the parameter of interest is the percentage or proportion of
heads. If the coin is fair, the probability of heads is 0.5. However, it is unlikely that many coin tosses would result in exactly 50% heads.
Because of the randomness inherent in the coin toss, even if the coin is fair there might be heads 48% of the time or 53% or even 37%.
A hypothesis test
is used to evaluate the difference between what we would expect if the parameter is equal to the specified value and what we observe in practice and to determine whether
observed difference is great enough to indicate that the specified value must not be the true value.
Use the applet to explore the logic of hypothesis tests. Click the buttons to toss the coin. The results of the coin tosses are displayed along with the probability of obtaining those results if the coin is fair. Based on the probability, determine whether you think the coin is fair.
In statistics, a hypothesis test formalizes a procedure for specifying a hypothesis and gathering data to decide whether or not
that hypothesis is true. The same basic method is used
to test hypotheses relevant to a variety of parameters.
The Hypothesis Testing Process:
- State hypotheses about the parameter.
- Collect data.
- Construct a test statistic.
- Apply a decision rule.
- Draw conclusions (in statistical terms and in context).
The next page addresses each of these steps in detail in the context of conducting a hypothesis test for a single mean. Subsequent pages show how to apply this process to a variety of situations including a hypothesis test for a single proportion, comparing two population means, comparing more than two means, and conducting hypothesis tests for categorical data. With a solid understanding the reasoning discussed in the next section, it will be natural to apply that same reasoning to all the hypothesis tests that follow.