V25 Hypothesis Testing

Welcome to part one of our video series in support of hypothesis testing. Okay, in this video, we are going to discuss what is hypothesis testing. We're going to discuss the two types of hypotheses, meaning the null versus alternative hypothesis, and we're going to review proof by contradiction. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, so, first, what is hypothesis testing? Okay, so, we just finished our chapter on estimation, okay, and we did estimation via confidence intervals, okay? But now, going forward, we're going to focus our attention on drawing conclusions or making decisions using what are known as hypothesis tests. Okay, however, confidence intervals and hypothesis tests are equivalent forms of statistical inference, okay, and we'll explore that more as we study hypothesis testing.

In a nutshell, a hypothesis test uses sample information, or information from the sample that you've drawn out of the population, such as… for example the sample average or the sample size, etc. Okay, and it's using this sample information to test some sort of a claim or a belief about a parameter value. Okay, so, we are trying to, with hypothesis testing for example, trying to test a belief or a claim, perhaps, about mu, or mu1 minus mu2 if we've got two… two populations, or perhaps P, the… the population proportion, or perhaps Sigma 1^2 to Sigma 2^2, right? These are all parameters, okay, and we're doing this to ultimately draw a conclusion about that parameter value.

Okay, there are two types of hypotheses. The first is known as the null hypothesis, okay, and we label that as H sub o. Okay, an example of a null hypothesis might look like this. Ho is, for example, let's say we want to test a belief about population mean, mu, equals some value that we believe that it is. Okay, so, maybe we think it's 25, or maybe we think it's 30. We don't know but this is the… our… our belief. Okay, it is… it's our existing belief or claim about a particular parameter value. Okay, so, again, this 30 is… it… it could be 3,5 it could be whatever our existing belief is about it. Okay, so, this existing belief can be viewed as… because it's existing the status quo, or perhaps it represents some historical knowledge about that parameter value.

Okay, in problems that we're going to do involving hypothesis testing, the null belief could be posed to you as, perhaps, some sort of advertised value about the parameter, or some sort of standard or specification value involving the parameter. Okay, so, these are all ways that you can view the hypothesized value in the null hypothesis. Okay, the alternative hypothesis is labeled H sub one, okay, or you may see it labeled as H Sub a for alternative. Okay, but, the alternative hypotheses to the above null hypothesis would… would be mu not equal to 30, okay, in this case. So, the alternative hypothesis directly opposes the null hypothesis. The alternative hypothesis is the new claim, or the belief, that we're testing, okay, sometimes called the research hypothesis because we're… we're trying to test it. There are two possible conclusions with our hypothesis. The first is that you're… you're able to reject the null and therefore accept the alternative hypothesis, or you fail to reject the null hypothesis. Okay, one of two possibilities.

Okay, if you reject the null hypothesis, or are able to reject the null hypothesis, it's because there's sufficient evidence in the data to do so. If you fail to reject the null hypothesis, it's because there's insufficient evidence in the data to do so, and, of course, that's the sample data, right? That's the data that we have access to. Okay, now, if you fail to reject the null hypothesis, this does not mean you therefore accept the null hypothesis. So, if you fail to reject the null, this does not mean you therefore accept it. So, you should not say “I accept the null,” okay, and we will explore that a little bit more as we progress. But, you don't say I accept the null, and we'll learn why.

Okay, and the last topic I wanted to review is proof by contradiction, and you have likely seen this in one of your math classes. If not, it's very simple. Let's say your goal is to prove something to be true. Let's say we're trying to prove a to be true, okay, as we're often trying to do in math classes. Okay, proof by contradiction works via the following mechanism. So, if you're trying to prove a to be true, the first thing you do is to assume the opposite: assume a is false. Okay, you then proceed with the steps of your proof. Okay, if, during your proof, you obtain a contradiction of some sort, okay, then that means that… that initial assumption of a false, which we made up at step one, cannot be a good or accurate assumption. Therefore, if this initial assumption isn't good or accurate, if the assumption that a is false is not good or accurate, therefore a must be true, you've essentially then proven a to be true, and we'll follow this up in class when we talk more about how hypothesis testing relates to proof by contradiction.

We wish to thank the National Science Foundation under Grant 233582 for supporting our work. Thank you for watching.