Hypothesis Testing

Suppose our null hypothesis is that the mean age of students = 22 and our alternative hypothesis is that the mean age is not equal to 22.

Suppose we take a sample and find that the sample mean age is 26.

We can deal with this in two main ways:

(1) What is the probability of getting this sample result if Ho is true? This is called the probability method and the probability is called the p-value.

(2) What value of the sample mean would convince me that Ho is true/false? This is called the critical value method.

Using (1), we can calculate the probabilty of getting a sample mean as extreme as this on both sides (ie >26 or <18) if the population mean is actually 22. Suppose this probability turns out to be large, eg 34%. We would say that getting a sample mean as extreme as this if the population mean is actually 22 is not very unusual at all so therefore we would say that we do not have sufficient evidence to reject Ho. However, if the probability of getting a sample mean as extreme as this if the population mean is actually 22 is very low, eg 3%, we would say that this is very unusual and therefore we would reject Ho.

We usually use 5% as a cut-off point, ie we usually run a 5% Type I error - so there is a 5% chance of rejecting Ho when it is in fact true. This is called the significance level of the test.

Using (2) we can determine the critical values of the sample mean, C1 and C2 (or the test statistic, eg Z1 and Z2) such that the probability of a result lower than C1 or higher than C2 is 5%. Then if our sample mean (or the test statistic) is in this critical range, we will reject Ho at the 5% level.