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Hypothesis Testing

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Hi, this tutorial covers hypothesis testing. Let's start by defining both hypothesis and hypothesis testing. So a hypothesis is a claim about a population parameter, then hypothesis testing is the standard procedure in statistics for testing claims about population parameters. So a hypothesis test is going to be kind of a standard statistical procedure that will help test a hypothesis.

All right, so let's take a look at some motivation here. So I've noticed that many math teachers I have worked with over the years are left-handed. After a quick internet search, I found that 10% of the US population is left-handed. My claim is that the proportion of math teachers that are left-handed is greater than 10%.

Now, to test this claim, I would need to obtain a relatively large sample, preferably selected randomly, of math teachers. I would then need to calculate the sample proportion of those who are left-handed, let's say, p hat equals 0.15. So I know that the population proportion is p equals 0.1, 0.10. And let's say that I did take the sample of randomly selected, large sample of randomly selected math teachers, and I found that the sample proportion was 0.15.

Now that you've obtained a sample statistic, a hypothesis test considers what is the probability that a sample statistic equal to or more extreme than found could come from a population with a certain parameter? So basically, a hypothesis test would tell me how likely is this sample proportion to come from my population where the true proportion is 0.1. So is it likely that 0.15 would come from a population where the true proportion is actually 0.1?

Now, the first step in doing a hypothesis test is to write some hypotheses. So we always write hypotheses in pairs. We always write a null hypothesis and then an alternative hypothesis. So the null hypothesis, which we abbreviate is H0, is a claim about a particular value of a population parameter that serves as the starting assumption for a hypothesis test. The null hypothesis is always an equality statement, so it will always have an equal sign.

Now, an alternative hypothesis, or Ha, is a claim that a population parameter differs from the value claimed in the null hypothesis. The alternative hypothesis is always an inequality statement. So I have equality statements for the null hypothesis, inequality statements for the alternative hypothesis. The inequality symbols that we can use here are either the less than sign, the greater than sign, or the not equal to sign.

All right, so if we go back to the left-handed example, the hypotheses can be written as follows. So again, they always need to go in pairs. So my null hypothesis I'm going to write like that, and it always needs to be my parameter equals something. So in this case, my parameter is the proportion of math teachers that are left-handed. And what I'm going to do is set my null hypothesis to be 0.10. So I'll assume that the population of math teachers is the same as the population of just people in general, and that they each have the same left-handed proportion.

Now, my alternative hypothesis can be written, really, in three ways. I'm going to write all three ways down here. So it can be p less than 0.10, or p greater than 0.10, or p not equal to 0.10. So these are the three options that I have.

And in this case, since my sample proportion was greater than this hypothesized population proportion, I would probably want to pick this hypothesis. So the hypotheses that I would write here would be my null is p equals 0.10. My alternative would be p greater than 0.10. So these would be the hypotheses that I would set for my hypothesis test.

Now, once the hypotheses are written, statistics from the sample are used to make one of the two conclusions. And these are the only two conclusions you can make. You can either reject the null hypothesis, or you fail to reject the null hypothesis.

Notice, we're not using the language accept the null hypothesis. We're either going to reject it or fail to reject it. We'll never accept that it's true. Now, if we can reject the null hypothesis, that means we're going to have significant evidence for the alternative hypothesis. And if we fail to reject the null hypothesis, we do not have significant evidence for the alternative hypothesis.

All right, so kind of the-- more about hypothesis tests will come later, but this kind of served as your introduction. So this has been your tutorial for hypothesis testing. Thanks for watching.