 because we'll work from my screen for the table okay so I'm not going to start by explaining anything we just gonna go straight into the questions and start answering the question and then I will explain concepts and things as we go along so that we are all on the same page I have about in this session about 20 questions and I'm hoping we can do all of them so that it gives you an opportunity to practice hypothesis testing based on the theory that we have learned on hypothesis testing for the for the mean when the population standard deviation is known and hypothesis testing for the mean when the population standard deviation is unknown and hypothesis testing for the proportions and remember also when you answer question on hypothesis testing remember all those six steps of hypothesis we're going to rely on those steps when we do the questions and also when we get to the p-value and I need to also I think I included more questions that relates to the p-value how we find the p-value as well so that you get a practice in terms of remembering that when you find a p-value what you need to be doing with that okay so kick off some couple of questions we will do them together some of them I will let you do them alone as well okay here's the first question for testing the null hypothesis that the proportion is one over four or one quotas against the alternative which says the proportion is less than one over four it is given that the sample proportion is three over eleven and n is four hundred calculate the test statistic so yeah it's easy and straightforward all they want you to do is to calculate the test statistic since they have given us their proportions population proportion and sample proportion so it means our Z state is sample proportion minus population proportion divided by the square root of your population proportion times one minus population proportion divide so given the information we just need to substitute our sample proportion is three over eleven minus our population proportion is one over four divided by the square root of one over four times one minus one over four divide by our n is four hundred okay I could have just converted my three over eleven and one over four to decimals as well so you can do that and substitute the decimals do calculate if I convert three divided by eleven it's 0.272727 so I'm going to use my calculator on this one because it's easy I do the fraction and then I will add my fraction again as well and say three over eleven minus fraction one over four go down at the square root of a fraction and this way it gets tricky okay fraction which is one over four and use my arrow open bracket one minus over four I just complicated my life but you don't have to complicate it this way then go down and put the 400 so you just need to know how to use your calculator and what do you get from your site I will tell you what I get I got one one point zero four nine seven one point zero seven and that's the same as what you have so that if we run it to two decimal because our answer is two decimal then the answer is option two happiness let's see if this is if this was going to be a more complicated question sorry if you just go back you this is a one this would be a one-tailed test this is a one-tailed test because there's no not equal to sign in it yes so we always look at the alternative state see those are the kind of things that will stick in my head I would least remember that then I it makes it easy to know which direction to go yes next question calculate the P value of the test of the following hypothesis given that the sample proportion P equals 0.63 and equals hundred and the calculated test statistic is z is equals to zero comma zero five the null hypothesis and alternative are stated and they state this because they want to help us with finding the p-vane so the null hypothesis is equals to zero comma six zero and the alternative says it's greater than zero comma six zero since we can we need to find the p-value and we need to remember the following remember that the table the cumulative standardized probability table contains the probability of z less than a if we need to find the probability of z grace greater than a you remember we say one minus the probability of the value we find on the table remember that let remember that now I want to bring to your attention I think we did this previous we need to remember the following for a p-value if we need to find the p-value and number one if our z state is negative and our null hypothesis the science is greater than then p-value will be equals to the value you find on the table that's as straightforward as that if your z value is negative if it's negative and you use your z state to go find the p-value which is the probability on the table and the science says it's greater than the value you find on the table it is that value that you are looking for if your z value is positive and your alternative it says greater than then your p-value will be equals to one minus the table value the value you will find on the table that you need to always remember always always you need to remember that if z is negative and you need to find the probably the greater than a one-tailed apartheid p-value will be the value you find on the table it is positive and the alternative science is greater than then you're going to find the value by saying one minus the value you find on the table knowing that knowing what we are given needing to find the p-value and we told that it is greater than find the p-value so it means we need to go to the table we need to go and locate this value let's go to the z-table z-table we need to find zero comma zero five so we go here we do find zero comma zero five we also need to find oh not zero comma this is not the one no that's not the one we're looking for zero comma zero so zero comma zero at the top we're looking for five which is that so the answer is this which is zero comma five one nine nine so in order for us to find the p-value we need to say one minus zero comma five one nine nine and our p-value is zero comma four eight zero one and that will be option one. You know the other thing that puzzles me is identifying tables I don't want to lie how do we know which one like when you get a question like this which table do you go for? So how do you wipe with this? So for this question for proportion whether you're doing confidence interval or you're doing hypothesis testing any way where you see it says proportions you know that you're going to use z-table and that will be remember always that the z-table is what we call the normal table so let us just call it that the normal table which will be the cumulative standardized normal distribution table so which is this which is table E2. This is so somewhere on the on the page or the booklet or the past exam paper that we are using or anyway where you have the table maybe you need to say this is my z-table we need to name it that such and say this is my z-table but all the other tables and also there is a z on there on there as well just to show you what table it is. If you look at the t-table you will have a t a t and it will show you which table it is so you don't get confused. Next after this we're going to introduce another table so that is very complex. I'm using a question paper and I see the z that you're talking about oh yeah oh yeah thank you and then the other one says t-distribution okay no then thank you thank you. So when we introduce the next study unit we're going to use the chi-square table and that will be the only tables except the binomial and the poisson so because they are labeled like that binomial and then the other one is poisson so you just need to know them and and look at the key things on the table to know which table we're referring to. Thank you. So that is the PE value so let's go to the next question I want you to look through this question and then we will do it together from option one up until option five we'll do that together but I want you to look at it alone and do it alone and see if you can get the answer. I will give you five minutes just to help you when they ask you for the test statistic you know that we the population standard deviation is known I'll just give you the formula this will be the last time I'm doing it next time you will have to do it yourself so yeah because of the sign here it's greater than therefore our critical value we just use alpha so you're just going to find your critical value by using alpha so your five minutes has already started. Are we done? I see some responses. Remember we're looking for the incorrect antennae. Are we done? Two more minutes please. Lizzie that sounds like comfort food. That's peanuts. It's cashew nuts. Are we done? Let's continue. Got stuck okay so how did you substitute your X bar they are all given highlighted and indicated so it should be easy to substitute into the formula so it's four hundred and five minus four hundred divided by hundred divided by a spirit of one thousand and the answer you got is one point five eight one point five eight which is option number one yeah which that this is correct because it's one point five eight now the next question says we need to find the value so since the value looking at this it is greater than so it means we need to say one minus the value we find on the table remember that remember this information if it's positive and it's greater than one minus the value we find on the table and since this is positive so you go to the table you look for one point five which is nine four two nine so you will say one minus zero point nine four two nine and what do you get point zero five seven one we get point zero five seven one which is the same as the answer on the options which is not our incorrect one number three the critical value is zero comma zero five so we also doing a one take so it means we going to find the critical value by using Z alpha and therefore Z of zero comma zero five if we go to the Z table we need to look inside the table so it means we're going to go to the negative side because this side is the bigger number we're looking for zero comma zero five so not zero comma not that let me reduce the table a little bit zero two zero three zero four zero five so remember these are the two if we go up or go down it's five different so if I go up and I go here and I will notice that is that one that uses one comma four five one comma six four so it means our critical value for a one take critical value for one take is one comma six four five so what I'm closer to the time not now and later on you can also draw yourself a table you already have the Z alpha divided by two we need to do this you will have your one minus alpha which is your confidence level where we know that it is zero comma nine five our alpha is zero comma zero five therefore our alpha over two will be one comma nine six you can also draw for yourself only for the Z table this is doable not for the T table for the Z table only so for Z alpha we now know that it's one comma six four five and you can do for the other ones as well and then have this table as a reference table somewhere where you can always come back and refer to it especially when you do the Z table so that you don't always go to the normal distribution table to go find the critical value you can just use your table to find the critical values it will be a lifesaver if you can do it that way okay so that is just another useful information you can use or not use it's up to you so then this is incorrect number four and number five we can do it at one go because number four says the null hypothesis is not rejected the number five is the decision of that it says there is not enough evidence to reject because we know to reject in the null hypothesis to reject the null hypothesis at alpha level of significance of five percent so in order to do this we can either use the the p-value that we have and the alpha so if we use the p-value and alpha the rule say if I put the rule here it says if the p-value is less than alpha we reject the null hypothesis so now let's test that rule what is our p-value our p-value is zero comma zero five seven one and what is our alpha our alpha is zero comma zero five therefore our p-value is greater than and when it's greater than we do not reject the null hypothesis because there is not enough evidence to reject it so that is correct that is correct because step number four says we do not reject that's the correct answer and the step number five is just the decision that we're making that we're not rejecting the null hypothesis and that's how you should have answered it there is a question that looks exactly like this so I will expect you to do it on your own and get it right okay for example Lizzie can you just go one back quickly please as a belief okay thanks and I saw what I was looking for thank you okay for a sample of 35 items from a population for which the population standard deviation which is given is 20.5 the sample mean is 458 at 5% level of significance the tutor wants to test that the mean of 450 against the alternative which is the mean is not equals to 450 we are giving the sample we are giving the population standard deviation we are giving the sample mean we are told what alpha is and we just need to answer the question and find the incorrect statement we're given the alternative hypothesis which tells us this is a two sided test so it means if we're going to find the critical value we're going to find by using alpha divided by 2 when we make decisions there are two regions of rejection if we're going to find the p-value we're going I will explain it later on question or statement number one is that true or false true this test it is a two-tale test so it's true find the critical value z alpha divided by 2 which is z of 0,05 divided by 2 by now you should know what that critical value is that is 1.96 0,0250 if we go to the table that is the critical value we're looking for that's not the one this is the one which is 1.96 and if you would have done that table you would have stopped wasting your time from doing all this calculation and going to the table and then identified it from your summary table does it just show you ask is just showed on your table again for a moment I'm I'm not getting what you got there oh yeah that's a minus remember when we find the critical value we always just ignore the sign you just ignore the sign we will put the sign when we do the regions of rejection we will say this is the negative side and this is the positive side and if it's only a one-tailed table if it's a one-tailed side we only use the one side so always when you look at critical values especially when we deal with two-tale you regardless of whether it's the negative or positive we just use we ignore the sign only if they are deliberate and they tell you that it's a one-tailed test then you will need to put the sign because if if this let's say if the null hypothesis or the alternative hypothesis was greater than for this tail test this would have been 1,96 if it's less than it's minus 1,96 only for the less than for greater than it will be that if it's for equal we can just ignore the sign because it's a plus or minus so you must always remember that as well so those are the things that you need to remember for a two-tailed test we can ignore the sign in front because we know that it's both sides it's negative and positive on a one-tail if it's less than it will be minus 1,96 if it's greater than it will be 1,96 positive 1,96 so on this one it is a two-tailed test so we can just leave it as 1,96 okay step number three statement number three calculate the test statistic so now you need to go and calculate the test statistic that stat is equals to the sample mean minus the population mean divided by the standard deviation divided by the square root of n substitute the values our mean is 48 458 458 minus our population is 450 divided by the standard errors samples population standard deviation 20.5 divided by the square root of our n is 35 calculate and let me know what you get it's 2.31 458 minus 450 divided by another fraction 20.5 divided by the square root of 35 which is 2.31 which is the same as what we have number four it says find the p-value so we know that our test statistic is positive now when you find the p-value for a two-tailed test so finding the p-value for a two-tailed test we need to remember the following if our z is negative if our z is negative then we're going to say two times the table value and that will give us the p-value if z is positive then we're going to say two times one minus the table value and that will give us the p-value we have a positive value and we are doing a two-tailed so we're going to go to the positive side table look for 2.3 what did we get 2.31 going to find 2.31 which is 0.9896 so yeah we're going to say this should be because it's a two-tailed so there are two sides it's three times one minus zero comma nine eight nine six and that should give you two times 0.0208 you forgot 0.0208 yeah 0.0208 which is incorrect so our incorrect answer will be option four just to check also option five H naught is rejected at five percent level of significance we know what the rule say the rule says if the p-value is less than alpha we reject the null hypothesis so let's test that our p-value we found it it was 0 comma 0208 our alpha it's 0 comma 05 is this less than that yes it is we're going to reject the null hypothesis we can use that or we can use the critical value and the test statistic we can use the critical value and the test statistic because it's a two-tailed we have two regions of rejections one is minus one comma nine six and one is one comma nine six our test statistic is 2.31 it falls some way there in the rejection area so we reject the null hypothesis yes so whichever one you use you can make that same decision yes can I ask but didn't we we divide the alpha with two aren't we going to use that one use which one the critical value but the critical value we did find it it was 1.6 I mean in terms of the alpha when we have to test the p-value when we when we do the p-value no yeah since you get to think are we using the original value remember the decision says p-value and alpha not alpha over two we come to alpha yeah yes yes yes okay consider a one-sided upper tail so one-sided upper tail which means it's greater than so I want to go back to my reference the um what consider a one-sided upper tail hypothesis test with one percent level of significance which is our alpha of zero comma zero five um zero comma zero one and 20 degrees of freedom which this is our df our degrees of freedom if the population standard deviation is unknown and when the population standard deviation is unknown we use t what is the critical value so since we're doing an upper tail are we doing t alpha divided by two or t alpha which one t alpha we're doing t alpha so if we're doing t alpha and the degrees of freedom our t they told us our alpha is zero comma zero one and our degrees of freedom is 20 so go find the critical value on the t table zero comma zero one at the top so you will go find zero comma zero one at the top and then go find the degrees of freedom on the side you need to go find 20 so let's go to the t table t zero comma zero one at the top we'll look for zero comma zero one so we're looking for t zero comma zero one and 20 did you find it it's two point five to eight zero comma zero one and 20 which is two point two point five two eight let's go our answer is in the two decimal so it will be two point so our answer will be option number e Lizzie can you please just go back to your table again and just zoom out a little um I think I have I am having either the wrong table just okay I just like that critical values yeah you see it's like my head saying which table to use some tables they are called t distribution when you look at the table it should it should have a t somewhere there or they should have a t somewhere on there to tell you that it's a t table you found it hold on I'm checking I think I did yeah I was on the wrong table two point yeah I got it okay okay okay okay so I want you to try this one on your own the most important thing here is your not your your null hypothesis is your alternative hypothesis and the other information given so you need to go calculate your z and then use the z to go find the value on the table remember that it's greater than and I told you that we will have a question similar to that you have to do it on your own let me remind you just quickly moving back let me remind you if the p value is negative the value found on the table will be if the sorry if the z value is negative the value found on the table will be your p value if your z value is positive then you say one minus the value find on the table you must write this somewhere this for p value including also this for the two sided so that you always always remember that I'm going to grab water are we done I think the silence means we're still trying you want me to show you the table but I will need to know whether are you going to the positive or the negative side of the table for those who already calculated the z so let's and let's calculate z and then we can open the table and help those who don't have a table so what is our our mean the sample size n is 100 the mean and the standard deviation 3.4 and I don't know why they would have also asked you to calculate this the p value for is for this because if this is the sample standard deviation we cannot calculate the p value but anyway let's see um we have Lizzie yes smaller the favor I know you're going to do the whole formula but on so this is where I usually get stuck with some of these stuff they throw a lot of values in there I sometimes struggle to know which one it is for me everything you probably need to use but I know like your x bar what is the 3400 uh so yeah they say the mean and the standard deviation are found to be 3400 and 11000 respectively so because they said mean and so then this is the mean that is the standard deviation uh so the standard deviation will be 1100 100 divided by n n is 100 square root of 100 yes it's supposed to be 33000 minus 34000 no the sample mean this is your sample mean this is your standard sample standard deviation that's why I was saying this question probably they forgot to say yeah with the population standard deviation of this because we cannot find the p value for a t test remember your population mean is always given in the null hypothesis and it was given as a statement from the researcher the claim so our sample mean is 34 because it says to find whether the quickest door are exceeding the limit as sample of 100 resident is selected and the mean and the standard deviation of is there so I'm saying yeah they forgot to say population standard deviation in order for us to find the p value we need the population standard deviation not the sample standard deviation because when we find when we have the sample standard deviation we use t and with t we cannot find the p value it's not easy we can't find it but it's not easy to do it to calculate it it's a very complex calculation so what did you get yeah so it means you guys you didn't have seen this that was 0.909091 what what what what wait wait 34 was what am I doing 3400 minus 3000 no minus 3300 3300 not 3300 we can go fix that 3300 divide by 1000 divided by the square root of 100 which equals 0.90 so we can round it off to two decimal and we'll have 0.91 so now our answer is positive we're looking for the greater than so it means our p value will be equals to one minus the value we find on the table so one minus the value we're going to find on the table we need to go find z of 0.91 so we need to go to the z table and go find 0.91 which that is 0.8186 0.8186 and that should be 0.1814 0.1814 which is option two I want you to do this I'm not going to do it for you this one you will have to do it yourself I'm not going to give you the answer but I will tell you if it's correct or not correct so I'm also going to give you a hint on how to attack this question because this question if you look at the option automatically you can see that they are only asking you to do a conclusion make a decision and conclude now you need to read the question to see what you are given and what you will need to make this conclusion because there are two things that you can do to make a decision we can use the critical value you can use the critical value and the test statistic to make it to make a decision by using this if it's a one take or a two take we make those decisions or if the question what am I doing now so this is in case if it's z or it's t this will apply you will need to use the critical value so for t alpha so for if it's z you whether you're using z alpha or t t alpha or you're using z alpha divided by two or t alpha divided by two depending on the statement if you are using the p value alpha or let's use the vector alpha or you can use your p value or an alpha to make a decision and this is only applicable for z so based on the information given there are we going to use the critical value and the test statistic or the p value to make a decision let's read the statement a various literacy group recommend a reading of 135 watts per minute that's what the researcher is stating so 135 is our null hypothesis so because it says is equal then it means it should be a two-sided thing a teacher is convinced that the average reading for her class is different for the recommended reading speed in a sample of 35 so we are giving our n sample the average reading which is our mean x bar is 141 with the standard deviation so now they also say the sample has the mean and the standard deviation therefore this is s of 28 you are required to test the teacher's hypothesis that the class reading speed is different because also of a different means it's equal for the recommended reading speed at five percent they gave us our alpha five percent level of significance choose the incorrect statement below so since we know that population standard deviation is not given so sigma is unknown in this instance is unknown therefore we're going to use t test therefore it means we're going to rely on t not z so we're going to use t critic we're going to find t critical value and because it says different therefore it means we're going to have two regions of rejection so what you need to do so that goes as well what you need to do is two things you will need to go find the critical value t alpha divided by two you will need to go find t alpha divided by two which they gave you your alpha is zero comma zero five so you will need to go find t alpha divided by two and the degrees of freedom sorry i forgot the degrees of freedom and the degrees of freedom and find that critical value number two you need to go find z equals you need to go test calculate your test statistic z equals the mean divide by s divide by the square root of n you need to go substitute the value and calculate this so find those two make a decision and then we will talk you have five minutes ten minutes let me give you ten minutes so we will meet again at but if you got an answer let us know that you are done remember you can ask any question if you start when you need help are you winning almost there are we done hello are we done do you still need more time because you still have 30 more minutes it's still trying one minute are we done ten minutes gone go for it Leslie show us okay so i'm not going to do anything you're going to give me the information how do we find the critical value by using our level of significance of zero comma zero five divided by two on the t table with a degrees of freedom and minus one which is 34 so then you went and you use zero comma zero two five and your degrees of freedom of 34 yes so we gonna go to the table t table let me get a little bit bigger or smaller let's make it like that so our 34 and zero comma zero five which means it's this row this column and this row and you found two comma zero three two two yes so therefore our since this is a two take plus or minus two comma zero i forgot the numbers now three zero two two so it means our region of rejections those two two comma zero three i'm just only going to keep the two decimals it will be minus and positive two comma zero three for that one so those are our region of rejections so now calculating the z state your mean was 141 minus your standard deviation also your population mean was 135 divide by your standard deviation 28 28 yes so now let's make a decision where does 1.274 if this is zero in the middle where does it fall it falls somewhere in the white shaded area therefore it falls in there do not reject area so coming to before we make our decision we can look at the decision because the decisions are the first sentences like the first three weights that appear or four weights that appear here so the first one is do not reject so that is correct this one says reject so that won't be correct this one says there is no enough evidence we do have enough that's not correct we can make decision there we made a decision these says we reject the null hypothesis and conclude we do not reject so we left with two statements the last statement says we need to make a conclusion and when we make a conclusion we must base it on the null hypothesis the null hypothesis said there is a difference so it means if we are we if we are not rejecting so therefore the alternative will not have been true so the first one says we we reject the null hypothesis and conclude that the average is significantly different but we said it's not it's the there is no difference there so we say we're not rejecting so it means this statement will not be the right statement but this statement because it says there is a different there is not a difference so this is not significantly different will be the right one to use so your answer will be option number e so you would go back you would go back to this teacher and tell them what you're saying is not true yeah yeah no because we're not rejecting if we were rejecting it there would have been sufficient information to say there is they are different you understand we're not rejecting the null hypothesis that says the class reading is different from the recommended speed okay so it means they are not different you understand yeah yeah no no i understand it's a little bit tricky that way because the the statement says class reading is different from but we're not rejecting that so therefore it's not different yeah so that's actually what i that's what i was that's actually what i meant from a statistical point of view we do not reject the null hypothesis that that's good based on the the data we have but then you would say to her that the average reading speed is not different that same to her that she's wrong that's just how my mind is thinking at the moment okay yeah um other question consider the statements you given all the information number one is this a two-tailed test yeah yes yes it is a two-tailed test can find the critical value using a two-tailed test alpha is 0.05 is this statement correct yes the statement is correct because by now you should know that for alpha divide by two which is 0.025 the critical value is 1.96 calculate the test statistic substitute the value and calculate the test statistic what is your mean your mean is 53.5 minus your population mean is in the hypothesis testing which is 50 divide by your standard deviation is 10 divide by the square root of 64 calculate 2.8 is correct 2.8 will be correct now it's a two-tailed the value is the p value we need to find the p value it will be two times one minus the value you find on the table so go into the table 2.8 we need to go to the z table and go look for 2.8 9974 0.9974 which is two times one minus 0.9974 let's see again if I got it right yes I did what is the p value I'm getting 0.0052 0.0052 0.0052 therefore this is incorrect if we use the p value and alpha of 0.05 our p value is 0.0052 and our alpha is 0.05 is this more than that so no it is less than that so our p value of 0.05 is less than alpha therefore we reject the null hypothesis which means the statement is correct as well find the p value you first need to calculate your z-stat and use your z-stat to go find your p value your sample mean is 190 minus your population mean is 200 divide by your standard deviation is 50 divide by square root of your n is 9 calculate then go find the probability on the table remember the sign we know that the table contains less than value which are also called your p values what is your z-stat what did you get I got negative 0.6 negative 0.60 then take that negative 0.060 go to the table look for negative 0.60 that is your that is your p value which is 0.273 at one point do we ignore the the the signs what do you mean at one point do we ignore the sign we never ignore the sign after somewhere where you mentioned that we don't ignore the sign I don't know which one was that oh when we're talking about the critical value ah okay yeah the society is going to escalate because if you don't have if you didn't check with the sign if it's 0.60 the answer would be number two but yeah always remember that when we do hypothesis testing we always use the smaller side area so now if you get a probability of 0.7530 let's say it will not be right because it says you're looking for the bigger site you're looking for the bigger portion remember underneath the calf the area underneath the calf which is the sum of your probability should be equals to one so if it's a one-sided test let's say it's a one-sided test so if it's a one-sided test it will look like this so we only if it's uh let's use the less than site the less than site we're looking for this portion we're going to look for only this site everything in the bigger side so everything in this bigger site for the less than will be do not reject everything that is on this side will be reject so if your answer is this therefore it represents this site which is not correct you cannot make a decision based on the bigger site you need to minus one so that you get this portion to make your decision so you always always remember that if you go to the if your answer is going to be positive it gives you the bigger area let's go here you can see that it takes the bigger area we're looking for this smaller portion or if you lose if you use the negative site you can see we're looking for this smaller portion the negative site is easier because it contains the smaller area so any value you get the z value you get and it's positive you go into subtract it from one so that you can get the smaller portion so remember that to remember always so this is the tips are helpful thank you so that should always be at the beg of your mind okay professor of statistics refuses to claim that the average student spends three hours studying for the exam which of the following hypothesis is used to test this claim one two three four or five one two three or five i think it's two it is two is correct because the null hypothesis will say equal and the alternative will say it is not equal because they it just says they it claims that they spent three hours not more than not less than so just exactly so it will be a two tail and it will be not equal so this won't be even correct three you can look at it they have swapped in your null hypothesis we always state the equal side the equality and the null hypothesis they should never be an equal side yes sorry lizzie so the not equals will always be the one to the to the very right yes i'm just thinking of element yeah thinking of process of elimination it just makes our makes our chances it makes it easier yeah lizzie just go back again for me they please donkey okay state your null and alternative hypothesis one two three or four which one one two three four or five which one of it i think it's four i think it's two i'm also gonna go for four i will say you why four as well because it's uh the less than yes union doesn't believe it's uh it's less than 30 000 yes uh remember the claim yeah is not only what the statement is saying their way they say the average is 30 000 but they also say because the mean annual income of blue cola wakers in the country is less than and since this is the study about across the country is not about that company alone so we need to also take that into consideration and that says the mean income is less than so therefore it changes everything it means the population mean is less than remember this is the mean of the campaign not the not the country the country is the population so the mean the mean uh value will be the mean income will be less than 30 so if it's that i i could just even ignore what's on the null hypothesis and look at the alternative and i can see that all the alternatives won't be correct the only one that is correct is option four you can do that other you can start by by eliminating number one because it talks to the sample yeah that was that was what i'm gonna ask i i would assume in these hypothesis tastings you do not have this do you write the sample like that no this is incorrect