The reason for using an Inferential Test is to get a probability value, commonly called p-value. The p-value provides an estimate of how often we would get the obtained result by chance if in fact, the null hypothesis is true. In statistics a result is called statistically significant if it's unlikely to have occurred by chance alone. The most commonly used standard or cutoff is 0.05 or 5%. Because this standard, or cutoff is so important it has a special name. It's called the significance level of a test, and is usually denoted by the Greek letter alpha, so alpha equals 0.05. If the p-value is small, less than 0.05, this suggests that it is more than 95% likely that the association of interest would be present following repeated samples drawn from the population. AKA, a sampling distribution. If the p-value is less than alpha, which is usually 0.05, then the data we got is considered to be rare or surprising enough when the null hypothesis, H subscript 0 is true. And we say, that the data provides significant evidence against the null hypothesis. So, we reject the null hypothesis and accept the alternate hypothesis, H subscript a. If the p-value is greater than alpha, then the data is not considered to be surprising enough when the null hypothesis is true. And we say, that our data do not provide enough evidence to reject the null hypothesis. Or equivalently, that the data do not provide enough evidence to accept the alternate hypothesis. >> So, finding a p-value less than or equal to 0.05 means that the finding is statistically significant, and we can reject the null hypothesis and accept the alternate hypothesis. >> This p-value is also known as the Type One Error Rate, since it denotes the number of times we would be wrong to reject the null hypothesis when it was true. Rejecting the null hypothesis when it's true is also called a Type One Error. Looking at the p-value in our example, we see that there is not adequate evidence to reject the null hypothesis because the p-value was 0.17, which is definitely greater than 0.05. In other words, we did not reject the null hypothesis that there is no association between depression and number of cigarettes smoked among daily young adult smokers. We accept the null hypothesis. There is no association between smoking and depression, because the data does not provide enough evidence to accept the alternate hypothesis, that there is an association between smoking and depression. Let's change the research question slightly to demonstrate that the decisions you make about your sample and your variables can impact your findings and the conclusions that you draw. Using our example, we're still interested in the association between depression and smoking. However, we decide to not limit ourselves in considering only individuals who smoke daily. Let's look at a broader population of young adults, and consider those that have smoked at all in the past year, whether daily or more irregularly. The size of the sample in the NESARC data set is 1,706. With this sample, we find that young adults with depression smoked an average of 351.7 cigarettes per month with a standard deviation of 300 cigarettes. Young adults without depression smoked an average of 313.5 cigarettes per month, with a standard deviation of 268.2 cigarettes. So the difference between quantity of cigarettes smoked among young adults who smoked in the past year with and without depression is 38.2 cigarettes per month, almost 2 packs. The p-value of this revised scenario is 0.0285, obviously less than 0.05. This means that the probability that we would get a difference of this size in the mean number of cigarettes smoked in a random sample of 1,706 participants is less than 3%, which is a p-value of less than 0.05. So in this case, we can reject the null hypothesis, and say that young adult smokers with depression smoke significantly more cigarettes per month than young adult smokers without depression. If we look again at the number line of probabilities, we can translate this finding in the following way. If we reject the null hypothesis and say that there is a difference between the average number of cigarettes smoked per month among young adults, with and without depression, we would be wrong fewer than 3 out of 100 times. We'd be correct more than 97% of the time. Based on the standards of science, this is a level of certainty that gives us confidence in saying that there's a significant association between smoking and depression among current young adult smokers.