How can the normal distribution be used to help us determine if a specific value is likely or unlikely?

In the previous module, Continuous Random Variables, we focused on uniform and exponential random variables. A more useful continuous random variable is the normal distribution. The normal distribution will allow us to draw conclusions about how likely it is to observe sample statistics based on assumptions about a population.

Let’s say we asked adults how many hours of sleep they get a night. Because this value could be any value within a range, it is a continuous random variable. It is also likely that the overall distribution of sleep times would be mound-shaped and symmetric or approximately normal in shape. If the population mean was 7 hours and the standard deviation was 1 hour, we can see that some values are more likely to be a response from a person (7  hours) and some values are less likely to be a response from a person (10 hours).

Knowing the properties of a normal distribution will help us calculate probabilities associated with sampling distributions, which is the key building block of inferential statistics.