Occasionally, the mean of the population is known (perhaps from a previous census). After drawing a sample from the population, it might be helpful to compare the mean of your sample to the mean of the population. If the means are not significantly different from each other, you could make a strong argument that your sample provides an adequate representation of the population. If, however, the mean of your sample is significantly different than the population, something may have gone wrong during the sampling process.
After selecting a random sample of 18 people from a very large population, you want to determine if the average age of the sample is representative of the average age of the population. From previous research, you know that the mean age of the population is 32.0. For your sample, the mean age was 28.0 and the unbiased standard deviation was 3.2. Is the mean age of your sample significantly different from the mean age in the population?
The two-tailed probability of the t-statistic is very small. Thus, we would conclude that the mean age of our sample is significantly less than the mean age of the population. This could be a serious problem because it suggests that some kind of age bias was inadvertently introduced into the sampling process. It would be prudent for the researcher to investigate the problem further.