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One-Tailed and Two-Tailed Tests

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Hi. This tutorial covers one-tailed and two-tailed tests. So let's start with an example here. Every Friday, I treat myself to a bag of my favorite type of chips. I recently noticed that there seems to be fewer chips in the bag. So let's investigate this.

So what I would do first is check out the nutritional facts, and see if there's an estimate for how many chips there should be in the bag. And looking at that, I can see up at the top there, serving size, about 10 chips. Servings per container, 6. So there seems to be about 60 chips in each bag.

So what I would need to do, if I wanted to investigate this, is I would need to take a sample of a bunch of bags, count up the number of chips in each bag, and then take an average. That would constitute a sample average. So I would use that, then, to get up my hypothesis test. Before actually performing it, we have to have a hypothesis or a claim in mind. So let's define mu, and write some hypotheses.

So what I want mu to be is mu is my parameter, and I want mu to be the average number of chips in a bag. And I want to make some hypotheses. So remember that hypotheses come in pairs. You need a null hypothesis and an alternate hypothesis.

So since our parameter is mu, we would write this as mu and mu. You always need the same parameter in your null and your alternate. Remember that your null hypothesis is always an equality statement. So what I'm going to do is I'm going to assume that the average number of chips in the bag is what the nutritional facts say. So the average number is 60. So I'm going to assume mu is equal to 60. And we're talking about 60 chips here.

Now, the claim that I have in mind is, remember that I've recently noticed that there seems to be fewer chips in the bag. So if I have a claim in mind, that claim I have in mind is that mu is less than 60. So, remember, your alternate hypothesis or your alternative hypothesis needs to be an inequality statement. So I picked less than because that's the claim I want to test.

So this type of test, a less than test, is what's called a left-tailed test. So let's define that. A left-tailed test is a hypothesis test performed if the population parameter is suspected to be less than the assumed parameter of the null hypothesis. So in our case, when we said mu is less than 60 is our alternate hypothesis, that would be a left-tailed test.

In other cases, we might want to do a right-tailed test. So a right-tailed test is a hypothesis test performed if the population parameter is suspected to be greater than the assumed parameter of the null hypothesis. So if I was doing this-- in this situation, if I were doing a right-tailed test, my alternate hypothesis would be that mu would be greater than 60. So if I think they're over-filling those bags so they've given more than 60 chips, I would want to test this alternate hypothesis, which would be a right-tailed test.

So, then, a one-tailed test is just either a left-tailed or a right-tailed test. So both of these are types of one-tailed tests. So, to contrast that, a two-tailed test is a hypothesis test performed if there is no reason to think the parameter is higher or lower than the assumed parameter of the null hypothesis.

So if I were running a two-tailed test in this case, my alternate hypothesis would be that mu does not equal 60. So I'm not trying to show that mu is greater than 60. I'm not trying to show that mu is less than 60. I'm just trying to show that it's different. So two-tailed tests simply show a difference.

In two-tailed tests, extreme values above or below are evidence against the null hypothesis. So in this situation, if I got a sample mean of, say, 80 chips, that would give me evidence against the null hypothesis, and evidence kind of for the alternate hypothesis. Same thing if I were to get a sample mean of, say, 40. That would give me evidence against that null hypothesis also. So extreme values both above or below this assumed parameter will give you evidence against that null.

So one-tailed tests are more powerful than two-tailed tests. It's more powerful to say that something is greater than an assumed value, or something is less than an assumed value, rather than just different. So it's a stronger statement to say greater than or less than, rather than just different. So, because of that, one-tailed tests are generally preferred. That has been your tutorial on one-tailed and two-tailed tests. Thanks for watching.