 It's not a new chapter, welcome to your session 22. Today we continue with what we've been doing so this is just to assist you with preparing for doing your assignment 4 which is also due on the second but with the possibility of an extension. It might be extended by another week because assignment 2 and 3 are still open. They close on the second so assignment 4 might also be extended with another week but I always feel that you need to do your first submission as early as possible so that you have a feel of the type of questions that are asked so that then when you practice you already practice with your mind knowing what type of questions you will tackle when you look at your assignment as well and then use these activities and plus the recordings from the content and the activities that we did there to assist you with doing your assignment as well. So let's get to it so today I'm not going to discuss anything content we're just going to go straight into doing the activities the couple the first couple of activities we're going to do them together as a group who will answer the questions and then I will give you time as we go along so that you can also try and do it on your own and then we come back and do the feedback. I hope that is cool with everybody else so we'll first start with the activities that relates to confidence intervals. I do not want to combine the two chapters together because also I think when you write your assignment you first get the questions relating to confidence interval then later on you get the the rest of the other questions that relates to the other chapter that you're doing so they don't mix them they don't try to confuse you because also in the exam you do get it in that manner where they don't mix the questions up and around you get them in an order that they appear in your study guide as well so study unit one follows study unit two and like that so I did it that way so the first 10 questions or eight questions will be relating to confidence intervals and then we will do the other 10 or eight questions relating to hypothesis testing. I've sent you guys the table so for all the activities today we are going to use the table that I sent you which is called the STA 20 tutorial letter instruction to online exam. This is the the document that was sent to last year's students who were writing their exam so the table here has um oh this tutorial letter has several tables so we're going to use this table so the t-test on this one has three digits whereas the one that I was using on the tutorial letter 101 for 2020 had four digits so it creates some confusion because when when you're working with different types of tables or decimals when you drop decimal you are increasing the digits when you use the whole or four decimals the answer that you will get will be different to the one who worked with three decimals as one so to make sure that everybody or standardized everybody works on the same table same documentation then we're going to use this tutorial letter also when you go write the exam I'm hoping that the same tutorial letter will be sent to you but it will have 2021 information linked to it but the table will remain exactly the same it will not change so we're going to use this so week in week out from now until you go write the exam okay so that is which tutorial letter is this on um I've I've sent it on on what's up so I can I can also send it through the myunisa as well so this I shared it on what's up I will also share it via myunisa so that everybody else also who's not on what's up can also have the same information as well so let's get to it you said you sent it on the what's up group is it yes I got it I think I got it there is one person who also responded so you can just click on that response that yeah the question that so when you click they don't take you to the original PDF that I sent okay so since we're going to do this together you can we can all we will do two questions together and then I will leave you to do the rest of the questions on your own so a statistic practitioner to get a random sample of 50 observation from a population with the standard deviation of 25 and a sample mean of 100 the 95 percent confidence interval of the population is and we need to calculate or we need to find or construct a confidence interval what is it that we are given who can say what are we given on this in order for us to be able to answer this question we we given n of 50 what else population standard deviation 25 population standard deviation which is sigma which is 25 so therefore sigma is known and what do we know when sigma when sigma is known or population standard deviation known what do we use are we using z or are we going to use t to find the critical value use z going to use z and therefore we're going to find the critical value of z alpha divide by 2 what else are we given x bar 100 we're given x bar which is the mean of 100 and what else are we given 95 percent confidence interval so we're given 0.95 confidence level which is 1 minus alpha and what will be our alpha if we move alpha this side it becomes positive and we move 0.95 to the other side it's 1 minus 0 comma 9 5 and therefore our alpha will be 0 comma 0 5 and since our alpha is 0 comma 0 5 what do we need to do we need to go find the critical value which is z alpha divide by 2 just repeating this but you probably by now you know what our critical value is z alpha divide by 2 which will be z of 0 comma 0 2 5 0 what is our critical value yeah 1.96 that's correct so by now you now know that our critical value it's 1 comma 96 plus or minus if I can put it that way then since we need to construct a confidence level so we use the formula x bar plus or minus the critical value z alpha divide by 2 times the standard error which is the population standard deviation divide by n so our x bar you said it's 100 isn't it plus or minus your critical value you said it's 1 comma 9 6 your standard standard error population standard deviation 25 divide by the square root of n and is 50 do the calculation you can first do the calculation for the margin of error which let's see if we're doing it together I'll do it from here let's write all this so I have 1 comma 9 6 times into bracket fraction 25 divide by the square root of 50 with my error's close bracket equals 6 comma 9 2 9 6 I'm just gonna keep it to 4 4 decibels 100 plus or minus 6 comma 6 comma 9 2 9 6 9 2 9 6 I'm just gonna keep it at that then expand 100 minus 6 comma 9 2 9 6 and 100 plus 6 comma 9 2 9 6 did you do the calculations I'm lazy to do that I'm just gonna use the formula do you also get the same answer as I do yeah which is 93 comma 93 comma 0 7 if I look at the answer is in 2 decimals so I can also keep 2 decimals and if I go to the plus side you just add 100 so it's 100 106 comma 9 2 9 9 2 9 and 2 decimals it will be 93 therefore the answer is option number one happy is it just move your just move your calculates a little for the sake okay any question are we happy yes okay next I'm gonna give you time to look at the question and then we will do it together just read through it okay what are we giving on this question we're giving our mean oh sorry we're giving our mean bar which is 10 what else two variants of nine we're giving the sample variant of nine and what do we need to do what the sample variants find square root of nine you need to find the standard deviation which is the square root of nine which is equals to 3 what else are we giving sample size we're giving sample size of 16 and our confidence level of 0 comma 9 9 equals 1 minus alpha what will be the value of your alpha alpha will be 1 minus 0 comma 9 9 which is equals to 0 comma 0 1 now we need to find the critical value z alpha divided by 2 which is z of 0 comma 0 1 divided by 2 which is a z of 0 comma what is z of 0 comma 0 1 is a double zero double zero five and by two will be double zero five double zero five okay so what will be our critical value do you know remember I told you that you must do a summary table for yourself did you guys do that where you keep your your critical values because if you don't have that summary table that you created you will do the long route like me and go look for 0 comma 0 5 inside here which will be 0 comma 0 5 somewhere there and it's going to be 2.58 2.58 that is our critical value our critical value will be 2.58 so now we can go and calculate our confidence what's wrong with this all these things that we just did that's wrong do you know why it's wrong because is it not because what is it it's unknown because the population standard deviation is unknown so it means the way we went and found the critical value it was all wrong so we need to use t alpha divided by 2 and the degrees of freedom which is n minus 1 so which then t of 0 comma 0 1 divided by 2 and the degrees of freedom of our n is 16 minus 1 gives us t of what will be 0 comma 005 and our degrees of freedom is 15 so we need to pay attention to the statements given so now we need to go to the t table not that not the z table because we're using t we go to the t table we're looking for t of 0 comma 005 and the degrees of freedom of 15 0 comma 005 is the last column it's 2.947 and that is the value 2.947 2.947 that is our critical value so mean bar plus or minus t alpha divided by 2 and the degrees of freedom and s divided by the square root of n substitute the values and calculate so our x bar is 10 plus or minus our critical value we did go find it was 2.947 times our critical value 3 divided by the square root of n and 16 go ahead and calculate the margin of error I'm gonna leave it at 4 decimal I'm only going to run off when I get to the final answer so you can also keep it as you see it 2.21025 so that will be 10 plus or minus 2.21025 so you can split it and do the lower boundary first 2.21025 and then go to the upper boundary which is 10 plus 2.21025 doing the calculation and the answer rounded to 2 decimal 7.79 that's the 7.79 if I round it to 2 decimal and in the positive side we'll get we just ate 10 12.21 rounded off to 2 decimal we left with 12.21 12.21 which means option 3 is the right at good oh happy yeah so far I think listening we're happy when you doing this now it's your chance to do it yourself okay so we can do this one also together because this one is the proportion okay so let's go back let's go back to those questions so we know that for this one we were able to know what to use because the population standard deviation is known we use alpha the second one they gave us the sample variance so we knew that when this sample variance is given therefore the population standard deviation is unknown then we use t for proportions remember if they didn't give you the proportion sample proportion they will give you observations for the proportion satisfying that divide by the sample size will give you the sample proportion and for the proportion we always use z so you must always remember that so what are we given on this question we are given the population size which is n 25 which we can just ignore because it doesn't add any value in what we're going to be doing so we can just also ignore that part we are given the sample size n of 600 we are given our x observations from that sample so there is those who opposed were x x is equals to 120 therefore we should be able to calculate p which is 100 or x divide by n which will be 100 divide by or 120 because that's 120 120 divide by 600 0.2 0.2 that is our proportion we are also given 0.90 which is 1 minus alpha so our alpha is 0.10 so since we're doing proportion we're going to use z alpha divide by 2 which is z of 0.10 divide by 2 which is z of 0.05 now do you know what critical value is that it is one of those exceptions so we go to the z negative side of the table remember when we look for 0.05 we're going to find it between those two values and it will be 1.645 so I'm going to repeat again on the side somewhere you can do what we call 1 minus alpha which will be your 0.99 0.95 0.90 which are those that I know by now so we know that this alpha will be 0.01 this will be 0.05 this will be 0.10 and what else we know if we want alpha divided by 2 this will be 0.05 and this will be 0.025 and this will be 0.05 and we know that if we looking for z alpha divide by 2 then we are looking for for this what did we find for this one is 1.96 and this is 2.58 and for a 90 is 1.645 I am missing one of them I think it's the 98 okay so let's let's look at this one 0.98 it's 0.02 which will be 0.01 and the critical value will be 2.33 so that is the other one so you must also be able for when you go to the confidence levels you must be able to find the same so for a 0.05 where it's just alpha it will be 1.96 no I'm lying sorry my bad I'm very amelior it's 1.645 z alpha of 0.1 what am I doing sorry wait please don't shoot the messenger I'm trying to get this is that it will be 2.33 because that's where you will find the same critical value for z alpha 2.33 and for 0.05 where it's 0.05 it's 1.645 and for 0.10 we need to go and find that one 0.1 1 0 1 0 1 0 it's 2.33 then I got that one wrong then I got this one wrong oh am I oh sorry that was right that was right 2.33 this is 1 0 not 0 1 1 0 1 0 will be somewhere here 1 0 it will be that one which is 1.28 28 for 0.01 it will be 0.02 you need to find 0.02 0.022 anyway I'm not I'm not sure whether they will give you for a 0.98 but those are the those are the common ones you will you will find or you will use so this you will use now for confidence interval for confidence intervals and for hypothesis testing when it is not equal for this once you will use when it is greater than or less than for only for hypothesis testing so I'm already I've already did the summary table for you so I'm not going to go back to this so you must have it some way we're going to use it as a reference you don't have to always go to look up the values so I'm going to use this okay only for z only for z for for t we'll have to go and find the values so I'm deleting it I hope you wrote it down if you didn't write it you can pause the video when you get to this point and and write the the the the table okay that is too long now time to get you a table okay so now let's answer the question we are calculating proportions so this will be p plus or minus the formula plus or minus our critical value z alpha divided by 2 and p1 minus p which is our standard error in this instance we don't use the populations we use the sample because we have only the sample information substitute the value 0.0 plus or minus we are oh gosh now you need to tell me what critical value did we say for 0.05 1.645 1.645 times 0 comma 2 times 1 minus 0 comma 2 divide by our n of 600 close bracket I am just going to do the whole equation since I've used up most of the time so 0.2 I'm not going to start calculating the margin of error first I'm just going to go straight and do the whole calculations 6.645 times square root of fraction 0.2 times 1 minus 0.2 close bracket divide by 600 and I need to go close the bigger bracket okay then equal the on the lower side I get 0.1731 0.1731 I'm going to leave it to four decimals so go into the plus side I get 0.269 0.22 that is a 22 2269 just option one that will be option one so there is your exercise you can do this one on your own you want to give you five minutes identify what you are given find the critical values substitute use the right formula to substitute when you're done please let me know don't be quiet I just want to know how far you are and you can also post your answer on the chat go and submit I didn't post wait Slyzy I think I'm finished okay thanks I'll wait for the others um can I I'm allowed to post the answer yes you can thank you for posting the this I hope it helps can I see the answers please oh sorry I'm muted okay we're given all these values uh your critical value will be 1.96 1.96 I'm sorry I was writing but I was speaking and I muted myself okay sorry about that okay so now we need to calculate z alpha divided by 2 times the standard error which is the population standard deviation divided by the square root of n our mean it's 120 plus or minus our critical value 1.96 times standard deviation 20 divided by the square root of n what do you get did you get so the the the one after the the plus sign or the minus sign is 7 comma 1569 that that excludes the 120 if you're looking for that just the one after this the expected correct 120 plus or minus one you say 7 comma 1569 oh thank you 120 minus 7 comma 1569 and we do the upper side 120 plus 7 comma 1569 and the option will be 2 option 2 option 2 happiness the next question the sample of 25 they gave you the average time take it takes 30 minutes they just give you all that information and they also tell you that the population standard deviation is 20 find the critical value that's all what they want so are you using z or t critical value are you going to find t alpha divided by 2 or are you going to use z alpha divided by 2 t y t sigma is unknown what is that so you're going to use z because the sigma is known so remember that sigma is population parameters we use grid letters so that is your population standard deviation you're going to find z alpha divided by 2 and what will be your critical value this should be quick and easy because you have a reference table option 4 will be option 4 because this will be your z of 0 comma your alpha here will be 0 comma 1 0 and alpha of 0 comma 1 0 if we divide alpha by 2 we get 0 comma 0 5 and the critical value will be 1 comma 645 a random sample of 100 results in a sample mean of 100 the population standard deviation is known to be the 5 what is the margin of error for estimating a 99 percent confidence interval are we using z to find the critical value or are we using t are we going to use t alpha divided by 2 or are we going to use z alpha divided by 2 okay the standard population standard deviation is also known so it should be z it should be z so if that is z then find the standard error sorry the standard margin of error and do you remember what the margin of error is if i have the formula that looks like this you know this will be the formula that you get calculate the margin of error what what which part of the equation are you using or the part of the formula you're using to calculate this after plus minus you're going to use only that part then calculate the margin of error then are you done what is your critical value critical value value is 2.58 2.58 times your standard deviation 35 divided by the square root of 100 that gives you 9.03 2.58 times i'm just passing time times 35 divided by the square root of 100 it's 9.03 what are we given here we are given the observations so it's going to be the proportion it's going to be proportions because also in the question it says the proportion so what do you need to do is to find your proportion which is z alpha divided by 2 you need to find what your critical value is and you need to multiply by so what are we given in the question we're given our n you're given your p which is your proportion you don't have to go and calculate it you are given your 0.95 confidence level and you can go find the the proportion but read the question carefully it says they are looking for the upper limit and if we're looking for the upper limit which part do we live out which part do we remove minus we're going to remove the minus we're only interested in the plus side so you can do that calculation are we done yeah i'm done okay let's do the answer sorry say again Lizzie okay Lizzie you're still waiting for us so we're waiting for you i'm waiting for you i thought someone said just a bit so i was waiting for that person to to say they are done okay okay sorry okay so how do we answer this um our p is 0.48 plus our critical value 1.96 1.96 so even though they're asking us for the upper limit as long as it's confidence intervals we're going to divide alpha by two so it's 1.96 times the square root of 0.48 times 1 minus 0.48 divide by our n is 500 so what is your upper limit did you all get the same answer 0.48 plus 1.96 times the square root of 0.48 times 1 minus 0.48 divide by 500 do you all get the same 0.538 option number three yes hey the last last last last question and then we do hypothesis you are also giving the population standard deviation on this so it means you are going to use plus or minus z alpha divide by two the standard error so when you read your questions in order for you to differentiate whether you are calculating for the mean or for the proportion it is information given so for the mean you will be given things like standard deviation the mean value or the average then you know that you calculate in the mean and you will need to know for the mean you have two options when the population standard deviation is known and when the population standard deviation is unknown and today we didn't cover much in terms of when the population standard deviation is unknown let me know when you're done are you done yes i hope so how did you answer it if you hope so what is your mean your x bar x bar is 78 78 plus or minus your critical value of 1.96 1.96 and your standard deviation 21 divided by the square root of 300 and if we use the calculator 78 minus 1.96 times the square root of 21 divide by 300 equal Lizzie your square root is over the entire term it must just be over the 300 yeah the square root is nice gets it here i'm already having calculator problems i'm it's very personal for me at the moment i don't know why i'm struggling so much as my calculator i'm fighting with it happy monkey there we go and the answer is 75.62 and if i need to find the other value i'm just going to change the plus 80.38 which is option number two yes option two i'm so just calculated that as you're saying that's amazing it is you see how quickly it does everything yes for you okay so we're done with confidence intervals so by now you should have a feel in terms of the type of questions you can get and answer you should be able to answer any question relating to confidence interval with ease by now so let's look at hypothesis testing more questions i'm going to start with this question because i think previously i even cut it off the video then so that we don't have to come back to it on that last week's question so i've added it to this week's exercise as well because i think there was an error an error on the question and also because we did it in a hurry without explaining a lot of things so there was so much happening at the same time so i've included it again so with this the next hour we will do hypothesis testing so just to remember with hypothesis testing the sign that goes there doesn't matter we can have the less than or equal greater than or equal it means nothing or we can just have an equal sign there you must just remember that the most important sign that you always always need to make sure that it's correct it is the alternative because with the alternative sign it will tell you whether you're doing a one-tail test a two-tail test it will tell you where your original rejection will be on the left or on the right or whether if it's a two-tail whether it's on both so that when you make conclusion using the critical values you do not get confused otherwise if you are using the p-value you also need to know that if it's less than the value you find on the table will be the value of your critical value unless if your z-value it's a positive z value then you need to do one minus the positive value you find on the table and so forth so that sign is very very important so based on the question that was asked so the question says in a sample of 36 the sample mean is 84 it is also known that the population standard deviation is 16 you are required to use this information to test the hypothesis that the mean population is greater than or equals to 80 where the alternative will say it is less than okay so we need to find the incorrect statement in this what are we given the sample size n 36 what else we given the mean sample 84 we are also told that the population standard deviation which is sigma is 16 so therefore the population standard deviation is is known in this so therefore it means for anything that we do we're going to use the z-test statistic or we're going to find the z-critical value or we're going to use the the z-table for whatever question that might follow now the first statement they are asking you is based on this information that you know is this test a lower test test statistic tell you test statistic so if you can remember what's what just always draw yourself a picture and say this is my lower limit this is my upper limit so in this side on this side we're going to use the less than because our region of rejection will be going in the less that direction the arrow is facing that direction that is why it's the less than on this side the arrow is going that side so therefore it will be greater than on that side which is the upper side because we know that this is the lower because it will have the minus the site and we will have the plus the site way because we know that in the middle that is where zero is for our standardized normal distribution table so the question here is asking based on this information is this the lower or upper area or upper z test that's a question that i'm asking you guys it's the lowest level it will be the lower so therefore that will be correct uh then the next question is asking you to calculate the test statistics do you still remember how to find the test statistic your z-stat you need to calculate it which will be your sample mean minus your population mean divided by the standard error which is the population standard deviation divided by the square root of n substitute the value and calculate and let me know if you get it correct so our sample mean is 84 minus our population mean is always given in the question is 80 and our standard deviation divided by our standard deviation is 16 divided by the square root of our sample size is 36 what is the answer that you get i'm not going to do the calculation so the answer there is one comma five zero because we're going to be going to the z table i'm leaving it as two decimals option three is asking at 10 percent level of significance the rule is to reject the null hypothesis if the p value is less than or equals to zero covers one remember the decision says if p value is less if the p value is less than alpha we reject the null hypothesis that's the rule and that is what that is saying it just repeating the same rule is that the right rule is it correct that's whatever statement that they are saying does it contradict the rule that i just said it's the rule that you need to use when we make a decision it's correct it is correct because they all what they're doing there is just repeating the rule to say if our alpha which is 10 percent at this point we know what our alpha is if alpha is 10 percent i would and p value that we're going to get if it's going to be less than the value of alpha are we going to reject that and that is the rule that's correct the way they speak at the rule yes does the fact that it says less than or equal to does that not change the weight of the answer i'm just asking no it makes no it makes no difference no it makes no difference okay okay um i know that in some of the nodes actually i think they say when it is greater than or equal we reject or when because if it's at this if it's on the top yeah whether it goes this way we're still going to reject because it's the less than if it was greater than and it falls in this side we will not be rejecting something like that so it will not be oh sorry here i'm talking about the critical value not the p value but yes so um i don't think it will be a problem whether it's great and less than or equal and exactly but i know that at some point on the nodes i said if it's greater than or equal then we do not reject the null hypothesis but this is just for them to state in the rule that when it's less than or equal are we going to reject the null hypothesis it's just the rule number four um is asking you to find the p value so now we've calculated the z test you need to take that z test go find the p value on the table remember the rule that remember the information you know from the normal probability table that the table contains the less than values or even the sampling distribution where we set the table the cumulative normals uh cumulative standardize normal distribution table contains the probability of a less than so here we have a less than so it means we're going to find the p value which will be the probability of a less than value so we need to use that one comma it's positive so we need to go to the positive side and look for one comma five zero so we go to the positive we look for one comma five zero and since remember we're only looking for this margin the small area we're not interested in this shaded area because this zero comma nine three three two it is the whole shaded area but we only interested in the small portion of it so we're going to say one minus zero comma nine three three two in order for us to find the p value and what do you get so here you have to say your p value will be one minus zero comma nine three three three two what do you get point zero six six eight zero point zero six six eight this is the roman set which is the incorrect answer because what they did with this one if you take this multiply by two it will give you that answer it was if this was it would have been correct if it was a two tail equation then that would have been correct but that is the incorrect one number five it says at 10 percent significance level the null hypothesis is rejected so take your p value and take your 10 percent and check whether are we rejecting the null hypothesis so all what you do is you say zero comma zero six six eight and you've evaluated zero comma one zero is it bigger or less if the sign is less if zero comma zero six eight is less than zero comma one then we reject H naught and that's what that statement says is that correct and that is correct last time we met we had two options that were incorrect it was the upper here and it was the lower there so we cannot have two incorrect answers sorry I need to answer my phone moving on a laboratory tested a random sample of chicken eggs and found that the bean amount of cholesterol egg is 235 milligrams and the standard deviation is 20 milligram if the null hypothesis states that the mean is equals to 230 tested against the alternative hypothesis which states that the mean is not equals 238 5 percent level of significance with the assumption that the cholesterol of chicken egg is normally distributed and suppose that the test statistic is one comma 37 find the p value what do you need to do here what is it that you have to do you don't have to calculate the z steps they already calculated it they told you that it is z it is equals to one comma 37 all they want is for you to find the p value so in order for you to find the p value do you know what you need to do if it is a two take we did this last week Saturday if it's a two tell you multiplied by two you're going to multiply by two and since our z value is positive what do we need to do one minus the value we're going to find on the table so you just need to go to the table and go find one comma 37 on the positive side look for one comma three seven one comma three on the side and seven at the top and that will give you are we looking for one comma 37 one comma nine one four seven zero comma one nine four seven that's what we need to do so it will be two times one minus zero comma nine one four seven i hope i'm doing it right okay and what is the answer that you get it's option number five it will be two times one minus point nine one four seven which is zero comma zero comma one seven zero six just give me a sec okay let's check did i unmute myself okay so on the next one how many tissues tissues should the kimberlietlack cooperation package of tlinex contain the researcher determined that 60 tissues is the mean number of tissues during a cold suppose oh during a cold suppose a random sample of hundred tlinex users yield yielded the following data on the number of tissues used during a cold the mean average the sample mean was 52 this sample standard deviation was 22 suppose the alternative you wanted to test was that the mean is less than 60 if we know that state the correct rejection region for the alpha so all what they want you to do is based on this information if you have to make a decision the first thing you need to do is what will be your critical by looking at the options already i can see that they are using t but yeah they are also using z so we need to be very sure what which one we need so that we can eliminate some of these things so the first the first thing you also need to remember is the population standard deviation is it given the second thing where will be the region of rejection if it's a less than so you just need to make sure that to remind yourself to draw yourself a picture and write those less than or greater than and say where it is and if you forgot at the middle it's zero therefore anything this side it's negative anything this side is positive those are the things you always have to remember now since we're doing a one tail as well so in order for you to find the critical value do we use t alpha divided by two or do we use t alpha that's the one question that you need to also ask yourself are we doing t alpha divided by two or t alpha one tail is t alpha so we're doing t alpha so are we going to have one region of rejection which side the positive for the negative side the negative side the negative side so we can we can know that side so if it's t alpha we need to go find the degrees of freedom which is n minus one so what is our critical value our alpha they giving it to you they said it's zero comma zero five degrees of freedom what is our n oh it is one hundred ninety nine and is n is hundred minus one which means our degrees of freedom will be 99 so you need to go to the z the t table and go to 99 and let's look for zero comma zero five which is the second column one comma six zero so our critical value it's one comma six six how many sixes two sixes six six zero so yeah they used the four decimal table it's fine it gives us an idea so it means only those two options should be the right options so we know that that one will not be correct that one will not be correct and that one will not be correct also you must when you make a decision you must also make use of the sign so this one says it will be in the greater side and this one says it will be in the less than side which one is correct between the two because we know that our critical right so it's actually it's going to be in the negative side and therefore it will be option option two if you get a question like this in the exam will you know how to tag it it might not be like this they might ask you to make a conclusion or make a decision um so you just need to ask yourself follow up questions in terms of how do we get there to make a decision there are several things that you need to always remember the most important one is the alternative hypothesis testing the sign that says they tells you how you're going to make a decision because if you would have chosen number one you would have said it is in the site on the positive side so we know that it is in the negative side because it's a less than it's in the lower side and you need to know how to find the critical value because if you don't know how to find the critical value you might choose the wrong critical value as well the other important thing is when you find the critical value you must look at the information that you are given if it's especially when it's for the mean whether you're given the population standard deviation or whether you're given the sample standard deviation okay i'm gonna leave you to do this one on your own it should be quick and easy do you want me to go to the table for those who don't have not just reading the question oh sorry okay i'm done right to my needs okay do you want me to go to the question to the table silence means you all have everything you need what are we doing which what test are we doing a two tail or one tail single tail we're doing a one tail and we are also given what our z value and here we told our z value because we're doing a less than is negative two point four two if we go to the t desert table we go to the negative side table so i just want to remove all my markings we're looking for minus two point four two minus two point four two so when you go to the negative side you can see that we actually are looking at the smaller portion now so negative four two so what else do we need to do in order to confirm our answer what did we get zero comma zero seven eight zero comma zero seven eight is that the answer that we're looking for or is there something that we still need to do sorry lizzie um how do you tell again uh when it's a one tile or two tile alternative the sign on your alternative hypothesis tells you what you need to be doing so that's a if that's a not equals two then it's a two tail taste yes remember your alternative sign is the most important sign so we we know what our z value is we went to this table we found our table value my question is is this the final answer or is there something that we need to do 20 minus one from the answer but the table contains the z value of less than a value and we know if we want to find the probability we always find the probability will be the value we find on the table in this instance our z value is of a less than do we still need to minus because we're not looking for z for greater than only for greater than that's where we say one minus the table value so you need to always at the back of your mind remember the things that you've learned in chapters six of study unit six and study unit seven where we dealt with the table cumulative standardized normal distribution table for a one tail test where it's less than only when it's put when it's positive and I said this for a less than for a less than we are looking for that value and these values are those values when it's positive these values are the bigger side values but we're looking for the smaller one that's why we say one minus the value we find on the table in order for us to find the white area the smaller side on the negative side of the table it contains the less than which are the values that we are actually looking for so the probability you'll find here which is our p-value so this is probability when we were working with normal distribution chapter six chapter seven we were talking about probabilities in hypothesis testing we took this at what we call our p-values which are probability values p-values so it's just the name acronym that they use for them so if we are given a one tail test for a less than in an alternative hypothesis and we are given a negative z test we need to find the p-value which we can represent it as z of less than negative 2.42 if we go to the table and look for negative 2.42 we find the value the value we find on the table that will be the value we are looking for so and we did this last week I'm going to repeat it again so the answer is 0.78 but for interest sake let's just want to erase all of this so for the hypothesis alternative I'm only going to concentrate on the alternative if the sign is in the alternative if the sign is not equal if the sign is not equal and we need to find the p-value and the z value now I have the z value if the z value is positive we're going to let's start with the negative if the z value is negative and if the z value is positive so for a two tail so this is a two tail hypothesis test if the value is positive then we say two times the table value if it's negative sorry when it's negative we say two times the table value if the value is positive we say two times one minus the table value because we're going to the bigger side of the table the bigger portion for a less than let's start with the greater than so that we get to the less than late for where the alternative hypothesis for where it's an alternative hypothesis and our z value is negative then we say one minus if the value is negative we just read the table value because anyway it will give us let's do it this way let's draw this on the z table so the negative side of the table contains this small side of this thing that's the negative side or maybe I should not draw it so that we can if Liz if you want to draw if you want to draw that graph you must put it on on top of for example use a negative use a positive columns yes that's what I want to do actually so let's use another color for it oh no I can use the same color so this on the table we look for in all negative values on the table it's always going to look for those values there so that is that but when we make a decision yet that's why we multiply by two because then we we're taking into consideration that area and that area so because that's why we do in two times because we take that small portion we might we do it we split it into two we go to the negative and the positive so the negative and the positive they share that on the positive side of the table so if we have this so that you do understand that so it takes the bigger portion we don't want the bigger portion we just want the smaller portion and that is why we say one minus the value on the table because all what we want is to get that area and that area that's all what we want we want to take this white shaded area and represent it on both sides on the negative side and on the positive side that is why it is this minus one which will give us the blue area oh sorry the white area the one minus which will be minusing this red area so this is the the red shaded area that is why we say one minus that shaded area so that we can just get the white area which this will be our white area on that bigger on this that's all what we want so the small areas is the white area so on the greater than the same thing when the value is negative it's fine because it goes to the negative side of the table but since it is negative on this side since it's negative on that side so we're looking for this area even though we we getting the area from the negative side of the table we can represent it from the positive side so our critical value will be on the positive side if the z value is positive then remember we're talking about the big area we're going to say one minus the value we find on the table which we only are interested in the white area so this is just that white area there so that is why we subtract the red area so that we can get the white area the value of the white area so that is what we are doing with this so for a less than already for the less than for the less than we know that this are the table value we don't have any problem with that but when it is less than but we go to the positive side the z value is positive so this is the same we still so this one the shaded area will be on this side for the negative sorry I don't know how to draw I've never been good with drawing with drawing even from school so on the if our z is positive if our z is positive then because our z value is positive it will still be representing the bigger shaded area but we're looking for the smaller shaded area so we're going to say one minus the table value as well but also we're going to be looking at that small area so our p value will be that remember the p value is the probability which is the area underneath the kev and the area underneath the kev we only interested in that small portion of it not the bigger portion of it because the table contains the bigger portion okay so that is our hypothesis testing questions this will be your last question let me just see how many more questions did we have but you can go and look so 15 16 17 you can do on your own as you practice so this will be the last question that we target for today Lizzie you must please stay by the question awesome that's a belief are we winning are we winning you want more time hello are we there yet we'll get there when we get there left with five minutes I'm sure you can continue okay let's do it together then you can do 14 to 17 on your own okay I'm not going to answer the question I'm going to do the step one up until step six of the hypothesis test and then you can go and answer the questions yourself so in the statement we are told that the manager reported that there is a decline in the difference between 78 so our step one is to state our null hypothesis which will state that the population proportion is equals to 78 percent but what the researcher wants to prove is a decline which is a less than we cannot put it in the null hypothesis but it will go into our alternative hypothesis so our population proportion will be less than 78 percent that's step number one done step number two as to state what we given we are given the sample proportion of 76 which is our p we are given our n and we are given alpha I've already stated those step number three is to state what we're doing it's a z test and this is a one-tailed test it's a one-tailed test and we're doing a z test step number four is to find the critical value let me see if I need the critical value for this question I don't necessarily need the critical value but if I was going to look for the critical value the critical value would have been z z alpha which is z of zero comma zero five and null of z of zero comma zero five is because now I can see the chat picture is why can't I see all the pictures okay where is that we said z alpha of zero comma zero five is one comma six four five so if if we were finding the critical value we would find it as one comma six four five that is step number four step number five is for us to calculate the test statistic and I did put the formula I don't know why did I remove it z stat of p minus pi divided by pi times one minus pi divided by this by n substituting the value our p is seventy six zero comma seven six minus zero comma seven eight divided by the square root of zero comma seven eight times one minus zero comma seven eight divide by n which is 200 did you calculate this what did you get if you didn't calculate it I can go and calculate it negative zero point six eight negative zero point six eight point seven six minus point seven eight divide by the square root of point seven eight times one my one minus point seven eight close bracket divide by 200 negative zero comma negative zero comma six eight we need to go find the p value since our alternative hypothesis says it's less than and we're using a negative finding a p value so my p value will be the table value to the negative side find negative zero comma six eight and eight is the second last column which is zero comma two four eight three zero comma zero comma two four eight three I need to make a decision making a decision I can use the p value and the alpha I know what alpha is let's see they gave us our alpha is zero comma five so if I put the alpha of zero comma five is this less than or equal so this is greater than therefore I can say safely say we do not reject the null hypothesis now I can come in answer all these questions that they post because it's easy and quicker for me because if I looked at the questions I can see that they ask him most of the things that I need to know about hypothesis testing I can do the hypothesis testing quick quick and then come and answer the question the first question was test that the test statistics is zero comma six you've calculated it that is correct the p value is zero comma two four eight two anyway so probably maybe the typing error with the last digit that is the correct answer see the alternative hypothesis is that the population proportion is not equals to zero comma eight but I know that they said it's a decline so it cannot be a two take because it says it's not equal therefore it means they say it is a two take g the null hypothesis is not rejected I just made that conclusion it says do not reject the null hypothesis we can conclude that is not significantly different from 78 because we're not rejecting it and that's how you answer the hypothesis questions quick and easy as long as you know your six steps of hypothesis testing you should be able to answer the hypothesis questions for example if you can hear already all these are six steps of hypothesis null hypothesis test statistic critical value p-value conclusion null hypothesis critical value test statistic p-value conclusion all of them are in all those options so do the have the six steps of hypothesis testing in regardless of whether you're doing for the mean or the proportion as long as you see the answers options like this quickly on the side you do the six steps you come here you answer the question quick and easy say critical value test statistic p-value making a conclusion say this one is asking if you're doing a two-tailed test critical value test statistic p-value make a conclusion all those steps because you're null hypothesis an alternative hypothesis stated already there null hypothesis alternative test statistic critical value make a conclusion all the six steps of hypothesis done in one question and that concludes our time together I'm not sure if I want to see you on Wednesday I want to give you time I think for the next two weeks not two weeks today is let's go to the calendar so your assignment was due on the second which then gives you the Wednesday and the Saturday to do the Monday assignment but there is a possibility that it will be extended so that those who are still catching up with their assignment can catch up but your assignment should be due probably by the ninth so they will extend it by a week so if you haven't received a communication it might go out next week so that you know that but those who haven't done their first submission of assignment four I will urge you to start looking at your assignment four do your first submission those who already did their first submission I think you are now left with one unless if they extend and give you additional third submission do not rush to go and do your third your second submission unless if they extend it and they give you another additional submission but keep safe that and then on that week of the fourth and fifth then you can do your assignment I'm going to cancel all the classes that we supposed to have on those days so next week Wednesday and Saturday I am going to give you time to catch up and work through your assignment four there is no need for us to hurry because you have the rest of August to do the last two chapters which we will do together as well the same way so I'm going to give you that that week to do your assignment do you also need the week of the fourth to do your assignment because on the week of the fourth we can start with the new chapters I need to hear from you I think we can start with the new chapters on the week of the fourth okay um no let let let us let us not because there are one chapter is very short so we will do okay I'm going to give you two weeks we can come online if you want for an hour if you have any question I will I will be online on those Saturdays and Wednesdays for an hour to answer any questions that you have we can use those sessions as a question and answer session to iron out any issues any doubt that you still have any challenges that you still have we can use those two weeks to do that so the next time I will see you for class will be on the 11 when we do kai when we do kai square test so we'll cover the kai square test and do the activity of kai square test on the 14th then we will do logistic regression oh sorry regression regression on the 18th and then we do the activities on regression on the 21st and then the 25th and 28th we will do activities relating to both of those chapters right agree yes yes yes so I'm going to give you time to to work through your assignment because we don't have to have to to to rush on this once uh sorry Lizzie yes are you seeing on Wednesday