okay so let's take a look at that so less than 10% so pi is less than 10% 0.1 remember I'm going to write 10% as the decimal equivalent that's the proportion pi is the symbol for population percentage some some snap books by the way you'll see P here it's okay you can use P you could say P is less than quite one I use PI and then this was the claim right this was the claim so I want to I want to go ahead and write the word claim next to this just to remind myself that that's what the person said right what did the person say okay so what would be the the opposite of the claim now I did write this if you notice I wrote the claim under H a because it's a less than claim it doesn't have an equal to part if you guys remember anything with an equal to part has to be the no and something without an equal to side is always an alternative hypothesis so probably I would I go with pi is equal to 10% as my null hypothesis yes you could write it as greater than or equal to if you wanted that is the true mathematical opposite of less than but most the time we would we would just write the null is equal to alright so this would be a left-tailed test notice my H a is a less than pointing to the left so this would be remember H a decides the test never the no not the claim always focus on the symbol in H a less then the left hand looks like an arrow pointing to the left so it's a left tailed test okay so I'm basically got a one population proportion left tailed test with categorical data right that's kind of that's kind of the how the information that you two know about the test all right now we have the sample data already but I need to check the assumptions is this sample data good enough to do this test a lot of times the answer is no a lot of sample data has a lot of bias and is very messed up and can lead to type 1 and type 2 errors unless we're very careful about is my sample data good enough to do this test well let's check the assumptions right the assumption and first assumption was was it a random sample or representative of the population this was not a random sample but it was a census of the 2015 semester so I'm going to go ahead and say I'm assuming that it will represent the population okay so I'm gonna I'm gonna say it does pass the random sample or representative now are the individuals within the sample independent of each other that's a question that's one's a tricky one I'm not sure I don't think so just because we have certain stat students that are coming from the same classes we may even have certain stat students that are then her friends or carpooling with each other so that that's an interesting one probably not probably not but I'm going to go ahead and say let's assume that it did pass independence for now just so we can kind of do the test and gonna get a little bit of a practice with this now what about at least 10 successes well the number of successes is how many people have the characteristic of the percentage you're looking at so I'm looking for the number of people that carpooled in that case it was 30 so this is gonna pass the at least 10 because there was 30 now how many failures did I have failures means the number of people that do not have the characteristic of the percentage you're testing so how many people did not carpool well it's sometimes you would seem to note that as n minus X so 332 minus 30 would be 302 so that would be the number of failures 302 students did not carpool so that's going to definitely be bigger than 10 as well so the only one I'm worried about is the independence when that one might might not be true but I'm going to assume that it is independent for now just so we'll kind of continue with the test here okay so we're going to assume that this data actually met all the assumptions kind of assumed that this data met all the assumptions okay all right so we move on right I'm going to go ahead and calculate the test statistic critical value and p-value now again normally what you do is stick your sample data and your known alternative hypothesis into a computer program and it's going to sort of give you everything but I don't want to just give you an idea of how the test statistic would get calculated and the and the p-value and critical values so think about it this way let's say we said our earlier we're using a z-score test statistic the formula is P hat sample proportion - pi population proportion divided by the standard error usually a one proportion test statistic for proportion or means is usually the sample minus the parameter divided by the standard error but if you notice right here the standard error looks a little bit different than when we did confidence intervals I don't know if you guys see that see the difference but the difference is in confidence intervals we used PI P hats are B's P hat so it was square root of P Hat 1 minus P hat over N but now we're using we're assuming the null hypothesis is true right that's going to be the basis of a lot of our calculations so we're going to use that number instead of P hat so I'm going to use point one in my standard error calculation so I'm using square root of pi 1 minus PI over N or some stat books you would see that as P 1 minus P over N and then take the square root all right so basically plugging in the numbers we get point zero nine zero three six minus point one so the sample proportion minus the number in the null hypothesis and then I'm going to divide by the standard error square root of point one times one minus point one divided by 32 okay and if you work that out that comes out to negative point five eight five negative point five eight five four a couple things notice the negative that means my sample proportion P hat was definitely lower than my population parameter proportion in the null hypothesis and you can see they are this wouldn't the P hat is lower right so that goes with a negative z-score test statistic but this is not telling me how many percentage difference they are okay this is not a z-score test it says it does not tell you percentage difference it tells you the number of standard errors different okay so the sample proportion is 0.5 85 standard errors lower or below the population proportion of point 1 that would be the sentence if you're trying to describe this negative point five eight five by the way notice I didn't say negative point five eight five standard errors lower okay I'm different again the negative tells you it's lower than that the point five eight five tells you how much lower all right so that would be the sentence you would write now here's the key question right what about the critical value and the p-value and all that I mean does is this significant I don't know if you guys remember but when we studied normal normal calculations normal uh we found that you had to be about usually about two standard deviations away to be considered unusual or significant so usually this is not going to be significant usually it's got to get close to like negative two or something or at least enough getting close to negative two may not be all way to negative two but this is not even negative one you're like negative 1/2 this is actually a very not significant test statistic but we can see that better if we calculate the the critical value so again I went to stack he and I went to theoretical distributions normal just like we've done in the past and we're looking up the critical value right we talked about I did a video earlier about how to calculate critical values for hypothesis tests and and you can kind of see what we want remember this would be a left-tailed test so I went to the normal theoretical distributions normal calculator in stack key I click left tail and then I put in the significance level point zero five into the tail so that's the proportion in the tail and this number right here the computer gave me negative 1.645 right I know what you're thinking shouldn't it be one month 1.96 for a 5 percent for a 95 percent confidence yes if it was a two tail right this is only a one tailed test so it's not going to be 1.96 it's actually negative 1.645 so the question would be where does my test statistic fall compared to the tail all right it's all about the tail so does that fall in the tail well where is our test statistic well negative 0.5 85 I always like to write out the numbers here on the bottom so here here's 0 here's negative 1.645 negative 3 1 2 3 well where's negative 0.5 85 that's gonna be about right here right there's my test statistic so it's what is it in the tail no it's not right it's close to zero actually it's not not in the tail and that means not significant right so that means my sample data does not significantly disagree with my null hypothesis my sample statistic P hat does not significantly disagree with PI point 1 in fact this is telling me they're kind of close all right they're not even one standard error away about half the standard error away now what about the p-value how do I calculate the p-value well again I'll stay on the same statcato theoretical distribution normal calculator again I'm going to put a member the the p-value traditionally would be the percentage in the tail course or tails corresponding to the statistic now so all I did was I went to the theoretical distribution normal calculator and I put in the test statistic negative 0.5 85 in the bottom number and then it calculated the percentage in that's in the left tail corresponding to that and I got a p-value of 0.279 it's about a twenty seven point nine percent p-value so is that a high pvalue or is that a low p-value remember a high p-value is higher than the significance level low p-value would be less than or equal to the significance level this one remembered turned it into a percentage first what would this be as a percentage that'd be what twenty seven point nine percent twenty seven point nine percent that's definitely a high p-value right this is a high p-value so it's higher than our significance level so if you guys remember that means that if the null hypothesis was true this could just be the sample data could just disagree because all samples disagree a little bit that's called sampling variability or random chance this data could disagree just because of sampling variability it's not really we we can't really rule out sampling variability as a factor that's actually very important okay whenever you whenever you look at a p-value you should have this question in your head could my data be different just because of sampling variability and this one tells me 27.9% a high p value tells me that it could just be sampling variability