The average time for all runners who finished the London Marathon in 2009 was 272.5002 minutes (272 minutes and about 30 seconds). We want to determine if the LonRun13Samp data set provides strong evidence that the participants in 2013 were faster or slower than those runners in 2009, versus the other possibility that there has been no change.2626While we could answer this question by examining the entire population data (LonRun13), we only consider the sample data (LonMar13Samp), which is more realistic since we rarely have access to population data. We simplify these three options into two competing hypotheses:
The average 26 mile run time was the same for 2009 and 2013.
The average 26 mile run time for 2013 was different than that of 2009.
We call the null hypothesis and the alternative hypothesis.
Null and alternative hypotheses
The null hypothesis () often represents either a sceptical perspective or a
claim to be tested. The alternative hypothesis () represents an alternative claim under
consideration and is often represented by a range of possible parameter values.
The null hypothesis often represents a sceptical position or a perspective of no difference. The alternative hypothesis often represents a new perspective, such as the possibility that there has been a change.
TIP: Hypothesis testing framework
The sceptic will not reject the null hypothesis (), unless the evidence in favour of the
alternative hypothesis () is so strong that she rejects in favour of .
The hypothesis testing framework is a very general tool, and we often use it without a second thought. If a person makes a somewhat unbelievable claim, we are initially sceptical. However, if there is sufficient evidence that supports the claim, we set aside our scepticism and reject the null hypothesis in favour of the alternative. The hallmarks of hypothesis testing are also found in our judicial system.
A court considers two possible claims about a defendant: she is either innocent or guilty. If we set these claims up in a hypothesis framework, which would be the null hypothesis and which the alternative?
Answer. The jury considers whether the evidence is so convincing (strong) that there is no reasonable doubt regarding the person’s guilt; in such a case, the jury rejects innocence (the null hypothesis) and concludes the defendant is guilty (alternative hypothesis). Jurors examine the evidence to see whether it convincingly shows a defendant is guilty. Even if the jurors leave unconvinced of guilt beyond a reasonable doubt, this does not mean they believe the defendant is innocent. This is also the case with hypothesis testing: even if we fail to reject the null hypothesis, we typically do not accept the null hypothesis as true. Failing to find strong evidence for the alternative hypothesis is not equivalent to accepting the null hypothesis.
In the example with the London Marathon, the null hypothesis represents no difference in the average time from 2009 to 2013. The alternative hypothesis represents something new or more interesting: there was a difference, either an increase or a decrease. These hypotheses can be described in mathematical notation using as the average run time for 2013:
where minutes (272 minutes and about 30 seconds) is the average 26 mile time for all runners in the 2009 London Marathon. Using this mathematical notation, the hypotheses can now be evaluated using statistical tools. We call 272.5002 the null value since it represents the value of the parameter if the null hypothesis is true. We will use the LonMar13Samp data set to evaluate the hypothesis test.