One-tailed and two-tailed tests

in the last video our alternative our null hypothesis the drug had no effect and our our alternative hypothesis 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 says the drug has an effect that it will not the mean with when you have the drug will not be the same thing as the population mean and then the null hypothesis no your mean with the drug is going to be the same thing as the population mean 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 what if both ver what have both been considered in effect we did something called a two-tailed test this is called a two-tailed test two-tailed test because frankly a super high response time if you had a response time that was more than three standard deviations that would have 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 have a one tailed test and the way we could have done that is we still could have had the null hypothesis we still could have had the null hypothesis be that the drug has no effect 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 for some reason we already had a you know 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 h1 sometimes they write it as H alternative either one either one is fine if you want to do a one tailed test you could say that the drug the drug lowers lowers response time 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 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 and 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 has 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 one at one point oh five seconds is three standard deviations below the mean is three standard deviations below the mean so if we look at it let me just redraw it with our new hypothesis test so this is the sampling distribution that's the sampling distribution it has a mean right over here at one point two seconds and the result we got was three standard deviations below the mean three standard deviations below the mean one two three 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 what in that case and that was the last video you'd look at both tails but here we're saying we only cares does the drug lower does the drug lower our response time and just like we did before you say okay 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 consider when we put when we set our alternative hypothesis like that that we think it lowers so the problem if if if if our null hypothesis is true the probability of getting of getting a result more extreme more extreme more extreme than 1.05 seconds now we are only considering this tail right over here let me 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 okay the means definitely not one point two seconds but in this case we care about means that are lower so a result so now we care about the probability of a result lower lower than one point zero five seconds that's the same thing as sampling from the sample of getting a sample from the sampling distribution that's more than three 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 tail test where we only care about one direction below the mean and if you look at the one tail 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 be 0.13% so this one right here is going to be zero point or sorry 0.15 percent or if you were express it as a decimal this is going to be point or 0.001 five so once again if you set up your hypothesis like this you would have said if your if your null hypothesis is correct there would have only been a point one five percent chance of getting a result lower than the result we got so we are this is practice that would be very unlikely so we will reject we will reject the null hypothesis and go and go with the alternative and in this situation your p-value your p-value is going to be the point zero zero one five