 This is a video about finding the critical region for a hypothesis test. First of all, there are some definitions that you need to know. The critical region is the set of values for which the null hypothesis will be rejected. And the critical value is the first value that you come to on entering the critical region. Imagine that you start out from the expected value of your test statistic and you move to the left or to the right along the number line. The critical value is the first number you come to that's inside the critical region. And bear in mind that if you have a two-tailed test, then there will be two critical values. Thirdly, the significance level of a hypothesis test, which is also called the size of a hypothesis test, is the probability of getting a value inside the critical region if the null hypothesis is actually true. Okay, let's look at some examples and see how these definitions work. First of all, suppose we have a random variable with the Poisson distribution. And our null hypothesis says that the expected number of events, lambda, is equal to 10. And suppose we're testing that against the alternative hypothesis that lambda is less than 10. So we'll end up rejecting the null hypothesis if we have a very small number of events. And let's aim for a significance level of 5% or less. Okay, well let's look at a probability table and consider the values of x one at a time. What if x were equal to zero? Would we end up rejecting the null hypothesis in that situation? Or the probability of getting x equals zero is equal to 0.0000 to four decimal places. And clearly that's a very tiny probability and in that case we certainly would end up rejecting the null hypothesis. So that means that zero is inside our critical region. What if x is equal to one? Would we end up rejecting the null hypothesis that time? Well here we'd be interested in the probability that x is less than or equal to one. And the table tells us that that probability is 0.0005. Again, that's much smaller than 5%. So this is another situation where we'd end up rejecting the null hypothesis because the probability is tiny. And that means that one is also inside our critical region. It's the same with two. The probability that x is less than or equal to two is 0.0028, which is smaller than 5%. And therefore if x were two we'd reject the null hypothesis and two is inside the critical region. Three is also inside the critical region because the probability there is 0.00103 and that's less than 5%. And four is inside the critical region because the probability that x is less than or equal to four is 0.0293 and that's less than 5%. But what about five? Well the probability that x is less than or equal to five is 0.0671 and that's greater than 5%. So we'd end up not rejecting the null hypothesis in that case. We wouldn't have enough evidence to reject the null hypothesis if the probability turns out to be large, greater than 5%. So what we've discovered is that in this situation the critical region is the numbers 0.1234 because those are the numbers for which we'd end up rejecting the null hypothesis. The critical value is four because if we start out from the expected number of events and we move towards zero four is the first number that we come to that's inside the critical region. And the actual significance level of this test is 0.0293 because the probability of getting a number that's inside the critical region, the probability of getting 0.1234 is just 0.0293. Okay now notice that we can work out these things very quickly indeed just by looking at the table if we know what to look for. All we really need to do is to scan down the right hand column headed up by lambda equals 10 until we find the number that's nearly equal to 0.05. We want the largest probability that's less than 0.05. So we look down this column until we get 0.0293 because that's still less than 0.05 and it's the largest value that we can get to without going over 0.05. Then we look along the row from there until we hit four and that shows us that the critical region is all the numbers up to four, 01234 and that the critical value is equal to four. Okay now let's look at a different example. We've still got a random variable with the Poisson distribution and the same null hypothesis that lambda is equal to 10. But this time let's imagine we're testing this against the alternative hypothesis that lambda is greater than 10. So this time we'd end up rejecting the null hypothesis if we get very large values of our random variable. We'll stick with trying to find a significance level as close as possible to 5%. Okay so we'll look at the table of probabilities again and let's see if we'd end up rejecting the null hypothesis for various different values of x. So if x turned out to be 20 would we end up rejecting the null hypothesis? Well if we got 20 we'd have to find the probability that x is greater than or equal to 20. And remember that will be the same as 1 minus the probability that x is less than or equal to 19. And we'd find that by looking on the row above x is equal to 20. And it tells us that the probability that x is less than or equal to 19 is 0.9965. If you subtract 0.9965 from 1 you'll see that it's 0.0035 and that's smaller than 5%. So we would end up rejecting the null hypothesis if we got x equals 20. What if we got x equals 19? Well again we'd want to work out the probability that x is greater than or equal to 19 and that's 1 minus the probability that x is less than or equal to 18. So we do the calculation 1 take away 0.9928 which is 0.0072. And again that's smaller than 5%. It's smaller than the level of significance that we want. So again we'd end up rejecting the null hypothesis. It's the same if x is equal to 18. There we do 1 take away 0.9857 which is 0.0143 and that's smaller than the level of significance that we want. It's smaller than 5% so we'd end up rejecting the null hypothesis that time. We'd also end up rejecting the null hypothesis if x is equal to 17 because 1 minus 0.9730 is 0.0270 and that's less than 5%. We'd reject the null hypothesis if x is equal to 16 because 1 minus 0.9513 is 0.0487 but we wouldn't end up rejecting the null hypothesis if x were equal to 15 because then we'd want to know the chance that x is greater than or equal to 15 which is 1 minus the probability that it's less than or equal to 14 which is 1 minus 0.9165 which is 0.0835 and that's more than 5% greater than the level of significance that we want. So what this shows is that the critical region in this case is the numbers 16, 17, 18 and higher because those are the numbers where we'd end up rejecting the null hypothesis. The critical value is 16 because that's the first number that we come to as we enter the critical region and this time the exact significance of the hypothesis test we need to work out by doing 1 take away 0.9513 which is 0.0487 so the exact significance of this particular hypothesis test is about 4.87%. Okay, now just as before there's actually a much quicker way of obtaining this answer if you know what to look for. What we have to do is to scan away up the right hand column until we reach the probability that's about 0.95. We want the smallest probability that's still greater than 0.95. So we scan up until we hit 0.9513 and then this is the crucial bit, you need to remember this, it's very important instead of heading along that row you head along the row below until you hit 16 and that tells you that the critical region is the numbers 16, 17 and above and the critical value is 16. You must remember that you have to look along the row that's 1 below and remember that's because when you want to know the probability that x is greater than or equal to 16 that's 1 minus the probability that x is less than or equal to 15. Okay, so those were both examples of one-tailed tests. Now let's look at some examples of two-tailed tests. So we'll stick the same random variable which has the Poisson distribution with parameter lambda and the same null hypothesis that lambda is equal to 10 but this time we'll use the alternative hypothesis that lambda isn't 10. So this time we could end up rejecting the null hypothesis either if we see a very small number of events or if we see a very large number of events and we'll stick with the same level of significance as before we're aiming for 5% as our level of significance. So let's look at the table. Now because we have a two-tailed test and we could either end up rejecting the null hypothesis if you have a very small value of x or a large value of x we need to split the probability of 5% the significance level between the lower tail and the upper tail. So we want 2.5% for small values and 2.5% for large values. So what we'll do is we'll scan down the right-hand column until we reach about 2.5% and we'll scan our way up the right-hand column until we reach about 97.5%. So scanning down we get to 0.0103 and then we can't go any further without going over 0.025 which shows us that the left-hand tail, the left-hand part of the critical region is the numbers 0, 1, 2 and 3 and then we'll scan our way up the right-hand column until we get to 0.9857 because that's the furthest we can go until we get to 0.975 and that shows us that the right-hand tail, the right-hand part of the critical region is the numbers 18, 19, 20 and so on. So here we can see that the critical region is 0123 together with 18, 19 and higher and the critical values here are 3 and 18. Okay, this time in order to work out the exact level of significance we need to do some calculations. The probability of getting 012 or 3 if the null hypothesis is true is 0.0103 and the probability of getting 18, 19, 20 or higher if the null hypothesis is true is one takeaway 0.9857. So the exact level of significance here is going to be 0.0103 plus whatever we get by subtracting 0.9857 from 1. The exact level of significance is going to be 0.0103 plus 1 minus 0.9857 which is 0.0246. Okay, let's look at one more example this time about the binomial distribution. This time suppose the null hypothesis is that P the probability of success is 0.3 and the alternative hypothesis is that P isn't 0.3 and this time let's aim for a level of significance up to 1%. Okay, well we need a table of probabilities. Now we're going to scan our way down the column headed by 0.3 until we hit the appropriate probability and again we need to split the probability between the two tails so we need to halve 1% and you'll see that that's 0.005 so we need to scan our way down the column headed by 0.3 until we reach 0.005 and in fact when we scan down we get to 0.0026 and then we can't go any further without going over 0.005 so that shows us that at that end that part of the critical region is the numbers 0123 and 4. Okay, the next thing is to scan up the same column and to see what number we're looking for we need to subtract 0.005 from 1 giving us the answer 0.995 so we're going to scan our way up the column headed by 0.3 until we reach 0.995 when we do that we get to 0.9976 before we can't go any further if we go any further we'll have gone below 0.995 and that's not allowed so that tells us that the upper part of the critical region is the numbers 21, 22 and above remember we have to go along the line one below the probability that I've circled so now we know that the critical region consists of the numbers 0123 and 4 together with the numbers 21, 22 and so on all the way up to 40 because that's the total number of trials so it's the maximum possible number of successes Okay now as before in order to calculate the exact level of significance we need to do a calculation the probability that we get 0123 or 4 is 0.0026 and the probability that we get from 21 up to 40 is 1 minus 0.9976 so we need to add those two together and that gives us the answer 0.005 and so that's the exact level of significance in this case now one more thing before we stop that level of significance is actually quite far away from our target value I said that we were aiming for a level of significance up to 1% but we've got a level of significance that's about 0.5% actually it's exactly 0.5% so let's consider a little twist on the question suppose that instead of aiming for a level of significance up to 1% we just try to get as close to 1% as we can so this time we will be allowed to go above 1% if that could get us closer to 1% okay well in this case we can do slightly better because notice that 0.9937 is actually closer to 0.995 than 0.9976 is okay it's only 0.0013 away rather than 0.0026 so if we say that the upper end the upper tail of the critical region is from 20 upwards 20, 21, 22 and so on then we can get closer to 1% this time the calculation will be 0.0026 plus 1 minus 0.9937 and that's 0.0089 so there's a lesson here if you're given a question where you're supposed to be finding a critical region and you're given a target level of significance you need to pay very careful attention to whether that's meant to be the maximum possible level of significance that you're allowed or whether you're just supposed to be finding as close to that level of significance as you can because in this question that we've just been looking at if 1% was the maximum allowed then you'd have one answer whereas if you're just trying to get as close to 1% as you could then you get a slightly different answer so that's something that you'll need to pay attention to okay that's pretty much the end of this video about finding the critical region for a hypothesis test here are the definitions that you need to remember the critical region is the set of values for which the null hypothesis will be rejected the critical value is the first value that you meet on entering the critical region but remember that in a two-tail test there'll be two critical values and finally the significance level of a hypothesis test which is also called the size of the hypothesis test is the probability of getting a value inside the critical region if the null hypothesis is true and the significance level can be quite puzzling sometimes because sometimes we try to pick a significance level before we start like 5% or 10% but then when we actually find out what the critical region is it turns out that the significance level isn't exactly what we were hoping for and we need to work out the actual level of significance of our test so it could be that we're hoping for 5% but it actually turns out to be 4.2% or something like that okay well now that is the end of my video about finding critical regions I hope you found it useful thank you very much for watching