Hi, and welcome to the lecture on hypothesis testing as part of the Statistical Inference Coursera class. My name is Brian Caffo and this class is co-taught by my co-instructors Jeff Leek and Roger Peng. We're all in the department of Biostatistics at the Johns Hopkins Bloomberg School of Public Health. Given our existing work on confidence intervals, hypothesis testing will be relatively easy to develop. Hypothesis testing is concerned with making decisions using data. Typically, hypothesis testing starts with a null hypothesis which is specified, that represents the status quo. It's usually labeled H0, pronounced H naught. The null hypothesis, H naught, is assumed to be true, and statistical evidence is required to reject it in favor of an alternative or so-called research hypothesis. Let's motivate hypothesis testing with a straightforward example. A respiratory disturbance index, RDI, of more than 30 events per hour is considered evidence of severe sleep disordered breathing. Suppose that in a sample of 100 overweight subjects with other risk factors for sleep disordered breathing in a sleep clinic, the mean RDI was 32 events per hour. So the sample mean RDI was 32 events per hour, with a standard deviation of 10 events per hour. We'd like to test whether or not the population mean for this population that we're drawing the sample from is equal to 30, our benchmark for severe sleep disordered breathing, or whether or not it's larger than 30. So we specify that as H0, mu equal to 30, and the alternative hypothesis, Ha as mu greater than 30. In our example, we specified Ha that mu was greater than 30. However it could've been less than, or not equal to, depending on the research hypothesis that we were interested in. In that application, we're specifically interested in whether the respiratory disturbance index for this population was larger than 30. Notice that the truth is either that H0 is true or Ha is true. We could either decide Ho or decide Ha. Therefore there's only four possibilities. If the truth is Ho, and we decide H0, we have simply correctly accepted the null hypothesis. If the truth is H0, and we decide Ha, we have made what is called a Type I error. In the version of hypothesis testing that we're going present, we're going to control the probability of a Type I error to be small. If the truth is Ha and we conclude Ha, we reject the null hypothesis, then we've correctly rejected the null. And if the truth is Ha, and we decide H0, we've made what is called a Type II error. The Type I error rate and the Type II error rate are related in the sense that as the Type I error rate increases, the Type II error rate decreases, and vice versa. We can illustrate this pretty easily by considering a court of law. In most courts, the null hypothesis is that the defendant is innocent until proven guilty. Here, rejecting the null hypothesis is to convict the defendant. We're going to require evidence and a standard on that evidence to reject the null hypothesis. If we set a very low standard, i.e., we don't require much in evidence to convict people, then we're going to increase the percentage of innocent people convicted, Type I errors in this case. However, we would also increase the percentage of guilty people convicted, which would be correctly rejecting the null hypothesis. If we set a very high standard, basically a person has to have a smoking gun in their hand, to convict them, then we would increase the percentage of innocent people let free, a good thing, correctly accepting the null. But we would also increase the percentage of guilty people let free, or so called, Type II errors. And this illustrates how the two, the Type I error rate and the Type II error rate, are related. Of course ideally, what you would like is to get better evidence for a given standard. And that's the idea of doing things like increasing the sample size. But before we discuss those sorts of issues, let's talk about how we conduct hypothesis tests.