One-tailed and two-tailed tests

In the last video, our null hypothesis was the drug had no effect. And our alternative hypothesis was that the drug just has an effect. We didn't say whether the drug would lower the response time or raise the response time. We just said the drug had an effect, that the mean when you have the drug will not be the same thing as the population mean. And then the null hypothesis says no, your mean with the drug's going to be the same thing as the population mean, it has no effect. In this situation where we're really just testing to see if it had an effect, whether an extreme positive effect, or an extreme negative effect, would have both been considered an effect. We did something called a two-tailed test. This is called eight two-tailed test. Because frankly, a super high response time, if you had a response time that was more than 3 standard deviations, that would've also made us likely to reject the null hypothesis. So we were dealing with kind of both tails. You could have done a similar type of hypothesis test with the same experiment where you only had a one-tailed test. And the way we could have done that is we still could have had the null hypothesis be that the drug has no effect. Or that the mean with the drug-- the mean, and maybe I could say the mean with the drug-- is still going to be 1.2 seconds, our mean response time. Now if we wanted to do a one-tailed test, but for some reason we already had maybe a view that this drug would lower response times, then our alternative hypothesis-- and just so you get familiar with different types of notation, some books or teachers will write the alternative hypothesis as H1, sometimes they write it as H alternative, either one is fine. If you want to do one-tailed test, you could say that the drug lowers response time. Or that the mean with the drug is less than 1.2 seconds. Now if you do a one-tailed test like this, what we're thinking about is, what we want to look at is, all right, we have our sampling distribution. Actually, I can just use the drawing that I had up here. You had your sampling distribution of the sample mean. We know what the mean of that was, it's 1.2 seconds, same as the population mean. We were able to estimate its standard deviation using our sample standard deviation, and that was reasonable because it had a sample size of greater than 30, so we can still kind of deal with a normal distribution for the sampling distribution. And using that we saw that the result, the sample mean that we got, the 1.05 seconds, is 3 standard deviations below the mean. So if we look at it-- let me just re-draw it with our new hypothesis test. So this is the sampling distribution. It has a mean right over here at 1.2 seconds. And the result we got was 3 standard deviations below the mean. 1, 2, 3 standard deviations below the mean. That was what our 1.05 seconds were. So when you set it up like this where you're not just saying that the drug has an effect-- in that case, and that was the last view, you'd look at both tails. But here we're saying we only care is does the drug lower our response time? And just like we did before, you say OK, let's say the drug doesn't lower our response time. If the drug doesn't lower our response time, what was the probability or what is the probability of getting a lowering this extreme or more extreme? So here it will only be one of the tails that we could consider when we set our alternative hypothesis like that, that we think it lowers. So if our null hypothesis is true, the probability of getting a result more extreme than 1.05 seconds, now we are only considering this tail right over here. Let me just put it this way. More extreme than 1.05 seconds, or let me say, lower. Because in the last video we cared about more extreme because even a really high result would have said, OK, the mean's definitely not 1.2 seconds. But in this case we care about means that are lower. So now we care about the probability of a result lower than 1.05 seconds. That's the same thing as sampling-- of getting a sample from the sampling distribution that's more than 3 standard deviations below the mean. And in this case, we're only going to consider the area in this one tail. So this right here would be a one-tailed test where we only care about one direction below the mean. If you look at the one-tailed test-- this area over here-- we saw last time that both of these areas combined are 0.3%. But if you're only considering one of these areas, if you're only considering this one over here it's going to be half of that, because the normal distribution is symmetric. So it's going to the 0.13%. So this one right here is going to be 0.15%, or if you express it as a decimal, this is going to be 0.0015. So once again, if you set up your hypotheses like this, you would have said, if your null hypothesis is correct, there would have only been a 0.15% chance of getting a result lower than the result we got. So that would be very unlikely, so we will reject the null hypothesis and go with the alternative. And in this situation your P-value is going to be the 0.0015.