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## Hypothesis Testing

As the name suggests, hypothesis testing tests a hypothesis. Suppose it had long been believed that the average age of the inhabitants of a small town was 35 years old. Our hypothesis would be that the average age of this city is still 35 years old. To make sure this is still true, we decide to take a sample of 10 people and find the average of those 10 people at age 42. Does this mean that the average population of this municipality has increased? Or is finding the average age of 10 randomly selected people a realistic result of sampling 10 people? This is the question that hypothesis testing solves.

Statistics are steeped in probability and hypothesis testing is no different. If we decide that the average age in this city is now over 35, we need to be open to the possibility that we could be wrong. That is, the city population may still be 35, when our sample data indicates that the population mean is higher. This is an error and we want the probability of this outcome to be low, so we set the probability of this happening to a low number, usually something like 5%. This means that if we can show that the probability of making this error is less than or equal to 5%, we are willing to conclude that the town’s population is now greater than 35, based on the sample mean of 42.

Two characteristics of the sample are important for its significance. The first is the sample mean. If the sample mean of 10 was 48 instead of the original 42, we can intuitively see that it would be easier to believe that the population mean has increased. If, on the other hand, the average of the sample was 36 years, it would be more difficult to believe that the average age of the population has changed. A sample mean of 36 from a population that has a mean of 35 is much more likely than getting a mean of 42 from the same population.

The second characteristic of the sample that affects its significance is its size. Instead of a sample size of 10, suppose we take a sample of 20 people and find this mean to be 42. Wouldn’t this sample be more significant than the sample size 10? Yes, this means that if the population has a mean of 35, it is more likely to get a sample mean of 42 from a sample size of 10 than it would be from a sample size sample size of 20, and is therefore more significant.

As I said at the beginning, hypothesis testing is counter-intuitive at first. Check out my website and get a clear, visual explanation.

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