 have a sense of questions and we can start with the session. Welcome everyone to another session where we do revision. So today I just want us to concentrate on hypothesis as well. So we might pick the whole two hours to do hypothesis and tomorrow we can do the last two sections that are left which it will be your chi-square and the regression where we use the template. Let me share my entire screen. So if you're voting can still use assignment for questions. So left, right. Describe hypothesis testing so that then we can remind ourselves of the concepts that we want or we need to learn in order for us to answer the question related to hypothesis testing. So in terms of hypothesis testing like with chi-square there are three sections that you need to remind yourself of. Right? It's hypothesis testing for the mean when the population standard deviation is known and for the mean when the population standard deviation is unknown therefore it means they have given you the sample standard deviation and for the proportion and for the proportion. With with all hypothesis testing there are several things that you need to always remember that you need to remember the six steps of hypothesis testing where step number one is to state your null hypothesis and your alternative hypothesis. There's always two things that you need to remember at the beginning to state your null hypothesis and your alternative hypothesis and that also helps in terms of determining what you need to do with the rest of the statements. How are you going to make a decision? How are you going to find the critical values? How how are you going to find the p-value as well? It has so many things bearing on that. So the second step that you also need to remember is to state whatever you are given in terms of your alpha. So you need to state what alpha value you are given because it's also going to help you in determining your critical value, your p-value, how you make a decision and so on. Your alpha value, your sample size, you give it and sometimes you need to determine whether you are given the population standard deviation or you are given the proportion and that is very important because by identifying these things as early as possible will guide you in terms of whether do you need to use the hypothesis for the mean or do you need to use the hypothesis for the proportion and based on that it will also help in terms of determining the formulas that you are going to be using. Step number three, you need to find the critical value. Remember with the critical value as well depending on whether the population standard deviation is given or whether you're using the proportion or you're doing a one-tail test or a two-tail test you need to be able to determine where you will locate your critical values. So those things are very important whether you're going to be using the t-tail test or you go or sorry the t-table or whether you're going to use the z-table whether you need to divide your alpha by two or not those are very important. Step number four, you need to actually there is another step that I missed earlier on before step number two is the step number three should be state the test statistic that you're going to be using. So whether you're using test statistics whether you're going to be using the z or the t and the step number four you will find the critical value based on that whether you're using a test statistic or a t-test statistic and step number five it is when you need to calculate your test statistic. Calculate your test statistics and this your test statistic and this is based on whether which z value you are calculating and step number six is to make make a decision and conclude. So those are the six steps that you need to remember and know you must also bear in mind that in the exam the options can be the six steps and I haven't even touched on the p-value as well so those are the six steps that you're going to use if you are using the critical value. If you are using the z value there are also those similar steps step number one up until step number three when you get to step number three you come to step number four and calculate the test statistic and then you find the p-value and that we will look at it as we go on with examples. So how do you state your null hypothesis and alternative hypothesis? You state your null hypothesis on the basis that your null hypothesis always contains an equal sign so with your null hypothesis there is always an equal sign so it can either be I'm just going to use this slide here on this side it can either be equal sorry your null hypothesis will be equal or less than or equal or greater than or equal or it can just be equal because in your null hypothesis there is always an equal sign your alternative does not have an equal sign and your alternative will be not equal greater than or less than that's how you will state your alternative so this will be your null hypothesis how you state your null hypothesis and this will be how you state your alternative and also remember that your hypothesis statement we always use the paramita paramita therefore it means you're going to use the mean or you're going to use the proportion that is how you state your null hypothesis so if it says the null hypothesis you will state it and say the mean is equals to 100 or the mean for your alternative it will say the mean is not equals to 100 you cannot use x bar you cannot use the sample statistics when you state your null hypothesis those are very important things to remember so when you state your null hypothesis and your alternative hypothesis it also comes with different things it tells you what type of a test you are doing so if it is a null equal therefore it means you are doing a two a two tail a two tail test if your alternative remember only your alternative statement is the one that we're going to be using so in your alternative if it says less than or greater than or less than this it means you're doing a one tail a one tail test it is very very important to remember that that in your alternative hypothesis if it says greater than or it says less than that's how you will find your original projection and how you will find your critical values and how you're going to make a decision okay so when you do hypothesis testing for the population let's assume that we do it for a one tail test so when we do for the one tail test i'm just gonna repeat all this on this side so for a one tail test therefore it means there are two things to remember here so it's either your alternative is less than or your alternative is greater than so if we do for a one tail test you go into state whether you have your alpha value of um 0 comma 0 5 or 0 comma 1 0 or whatever the alpha value they have given you you go into state that you go you're doing a z test because this is a test you're going to state the critical value so now when you go find the critical value for a one tail test regardless of which site it is so you will find your critical value by using that alpha that is where you will find your critical value if you are doing a two tail a two tail test which means there is a not equal you will also find all these other things that you needed to find because we're doing for populations than a deviation where it's known it is a z test for a two tail test you're going to find your critical value by using z alpha over two all right so when you calculate your z test statistic for the z it you just use your z stat which is your normal z score value calculation from the sampling distribution which will be your sample mean minus your population mean divide by the population standard deviation over the square root of n and that's how you will find the test statistic you're going to use your test statistic and the critical value so depending on your test right so let's assume that the first one that we're going to use is for the less than in order for you to find the to make a decision for the less than you go into draw a normal distribution calf that looks like this and because it's a less than therefore it means your region of rejection will be on this on the negative side so you're going to have your negative z alpha there if it is greater than let's say your alternative was a greater than then your region of rejection will be your z alpha if it's a two tail then your region of rejection will be on both sides it will be on the z alpha divided by two on the negative side and on the positive z alpha over two that's how you will find and make a decision so if your z that statistic falls in the white area you do not reject if it falls in the red area for a less than you reject if it's greater than if it falls in the red area shaded area you reject if it falls in the white area you do not reject for a two tail when it is not equal if it falls in either side the shaded area you reject if it falls in the not shaded area you do not reject and that's how you will find the or you will make a decision based on the information that you have just want to give you some space so I can have enough space on here okay so let's also look at when the population standard deviation is unknown therefore it means they have given you a sample standard deviation for a one tail test when it's less than and for a one tail test when it's greater than you're also going to follow the same format and when it is not equal for a two tail test so yeah you're going to state your degrees of freedom you're going to find your degrees of freedom which is n minus one because it's going to help us because if I've already found out what my n is I can calculate my degrees of freedom I'm going to test the hypothesis and say oh my test statistic yeah will be a t test or t test statistic that will be step number three step number four finding the critical value for a one tail test it will be t alpha and the degrees of freedom and for a two tail for a one tail test so for both of them it will be t alpha and the degrees of freedom you would have already calculated what the degrees of freedom is for a a two tail test it will be t alpha over two and the degrees of freedom right so that's how you will find your critical value to and then you will calculate your test statistic which is t stat which is the sample mean minus the population mean divided by the standard error yeah your standard error will be for the sample it will be s divided by the square root of n which is almost similar to what I'm doing making the decision you're going to look at the side in your alternative hypothesis if the sign is less than then you're going to follow the same your original projection will be on the minus t alpha and the degrees of freedom sorry my hand is beating out it will be t alpha and the degrees of freedom if it falls on the shaded area on the negative side then you if it falls on the left hand side then you you you reject your null hypothesis if it's on the greater than side you also do the same you're going to determine what your t alpha and the degrees of freedom is and if it's a two tail you're going to find on both sides nothing changes it's still going to be on the t alpha over two and the degrees of freedom and on the positive t alpha on over two and the degrees of freedom so you're going to make a decision based on the original projection based on whether you're doing a one tail test or a two tail test now when it comes to the proportion similar thing happens you're going to do for one tail whether it's on the left hand side or the right hand side lower side upper side or when it is not equal and when it is a one tail test because this is that so the only thing you're going to find is alpha and the n and the proportion and remember for the proportion if they didn't give you the things that you're going to stay there if they didn't give you the sample proportion you're going to find your sample proportion by using observation satisfying the outcomes of the sample divided by the sample size and that will give you the sample proportion you will define what your level of significance is and so on and here you're going to state number three you're going to say you are doing a set test because for the proportion it's also a set test also similarly to what we did previously when it's one tail whether it's on the left or on the right use that alpha and for two tail you will use that alpha over two and it means you're going to have two regions of rejection and your set test will be calculated using the proportions which is the sample proportion minus the population proportion divided by the standard error which is the square root of the population proportion one minus the population proportion divided by n and that will be the set test that you use for the proportion and the same story happens for a one tail if it's less than it will be on one side if it's greater than it will be on the greater than side if it's a two tail you will have two regions of rejection so it will still remain the same negative alpha z alpha positive z alpha and negative z alpha divided by two and positive z alpha divided by two on the other side and that is if you're using the critical value to make a decision what happens when you use a p-vehicle this is only applicable when you're only doing a set test but for the set test any test hypothesis test that you do and you are using a set test so therefore it means for the mean and for the proportion when the population standard deviation is known for the mean and for the proportion you're going to be using the p-vehicle to make a decision how then do you make a decision based on the p-vehicle so there are three scenarios that can happen especially when you are using the p-vehicle now we will start with the one tail when it is less than so you will do the first step you state your null hypothesis and your alternative the same way as we stated you will state what you are given alpha your n your standard deviation your by all um your sample proportion whatever you are given there that's step number two step number three you will state that it's a test statistic you skip the critical value because we don't use the critical value you go ahead and you calculate your test statistic and you find your z step by using the formula that we have been using so you will use that formula to get your z step now in order for you to go find the p-vehicle the p-vehicle it's also known as the probability right it is also known as the probability veil what we know from chapter four on the not chapter four chapter six or study unit six on probabilities you remember the same logic if your p-z the probability of a z value less than a value we find that probability on on the table if it's the probability of z greater than a value we subtract the value we find on the table from from one right so the same concept you know the same things that you still remember you're going to apply here the z test statistic is your z value looking at your alternative sign if your alternative sign says it is less than therefore you need to say if my z step you keep the two decimals regardless of whether it's in the negative or the positive remember the table has negative or positive you're going to find the value on the table so when it is less than therefore your p-value your p-value will be equals to the value you find on the table that is the same the same very same concept that we use right so that is if it your z your z value it's less than right in your alternative you had that the alternative is less than then the p-value will be the value on the table if it is greater than therefore the p-value would be one minus a table value if it is greater than if your sign says in your alternative says greater than then your p-value will be equals to one minus the table value regardless of how much the z state value is your p-value will be the value on the table now come the two-tail test when it is not equal when it's not equal there are two scenarios that you need to take into consideration when your z value is positive let's start with negative when your z value is negative when the answer you get when you calculate your test statistic if that answer is negative then the following scenario will happen your p-value will be equals to two times the value you find on the table only if your z value is negative if your z value is negative you are going to multiply the table value or the value you're going to find on the table by two or you're going to add the two that value you find on the table to by to itself again if your z value if your z value is positive let's make it equals to positive the same way as we made that other one is equals to positive then the following will happen your p-value will be equals to two times one minus the table value those are the things we remember it's a mouthful the hands I said just there on on Monday last week or when we had the session last time to say it's gonna take us wrong to remember all these things we will need a session on hypothesis on its own if you can remember all this then it will be as a b c or x and y and z or as easy as pz to answer any questions in the exact because this just summarized what you need to know about hypothesis testing on the summary note that we also shared on the drive if you go to study unit nine you will find all this other information that you need to make sure that you know including also like theory information that they will ask you in the exam that you need to know everything that I just explained remember that that in your alternative we only use the population parameter we only use an equal sign we don't use the sample parameters and in your alternative there is no equal sign and all those things that I just mentioned what I didn't mention is the type errors you just need to remember what the type error one is is to reject them through null hypothesis and a type two error is when we we fail to reject a false null hypothesis and so on and all the things are here all the steps the six steps that I just went through with you there are the six steps you just need to know them the how to find the critical values we just went through this are those pictures that I just through just now in terms of how do you state your null hypothesis remember I said you can either use the greater than or equal or you can just use an equal side what this important is the sign you put in your alternative because it tells you whether are you doing a one-tail test or a two-tail test and it also guides you in terms of where is the origin of rejection if it's greater than the origin of rejection will be on the upper take which is the left hand side and this is just giving you an example of how you do the hypothesis testing in terms of the p-value which is something that I didn't talk about I just gave you an example of how you make how you find your p-value but remember that once you have your p-value you're going to make a decision based on your p-value and your level of significance which is your alpha if your p-value is small you reject the null hypothesis if your p-value is less than your level of significance you reject if it's greater than or equal it's exactly you do not reject the null hypothesis and that's how you make a decision based on the p-value and here is an example of how do you find the p-value if your if your z is negative and it is a two-tail test that whether it's a negative or positive sorry that's the other thing that I didn't cover on did I oh I did I did I did if it's negative it's just the table value times two if it's positive then we say two times the table value and remember this is only for when it is a two-tail oh yes I did cover that when it is a two-tail we just take the value we find on the table we add them together or we multiply that value by true to get to the p-value and then you just make a decision and this is just additional information so you will just follow through all this document to help you so if you have this and you use this in your exam it will make your life easier so let's answer the questions and answer the questions based on that information unless if there is a question is there a question do you have a not sure now Lizzie but they will be okay only guys can I get the summary okay they are on the site we we published I will share the link again on the chat later on okay so let's answer the questions in a sample of 36 the sample mean is 85 it is also known that the population standard deviation is 60 you are required to use this information to test the hypothesis where they gave you that the null hypothesis states that the mean give me a second somebody is not muted please make sure that you are future Elizabeth the other Elizabeth not me so that so they gave you the hypothesis the null hypothesis states that the mean is less than or equals to 80 and the alternative states that the mean is greater than 80 so based on the information just that we just learned no no no not so long ago not so long ago this information you should be able to identify certain things that be able to tell yourself what is it that you are given in this question so for example already here they're telling us the sample it's that step number two stating where what is it that you are given sample size n of 36 the sample mean of 85 the popular what is very important is that weight population standard deviation is 16 therefore our sigma is known and when it is known i'm gonna use the z the z test so those are the things that you need to think about based on their six steps remember all those six steps so if it is known then i'm using z test i'm giving all this other information and you are able to and state which one the null hypothesis and alternative they've stated it can i identify which one am i doing a one tail or am i doing a two tail so looking at you ignore the first part because that has nothing to do with what we need to state we look at that compact which is your alternative you look at the sign and it says less than oh sorry it says greater than therefore it means i'm doing a one tail test and it is an upper tail test right because it's on the right hand side then in my mind i need to think or in terms of a decision if i'm going to make a decision it means my original rejection will be on this side because i'm doing z z alpha because it's on the one side so as you can see i'm already thinking before i start answering the question about those six steps including all the other additional information that i will require so it will make my life easier to answer the question and find out which one of these statement is incorrect looking at this i'm assuming i'm going to be answering all six steps let's see question a or option a it says the test statistic is so here i'm just going to ask you to calculate what the test statistic is so we're going to use the formula as a z-spect is your sample mean minus the population mean the population mean is always given in your hypothesis statement divide by the population standard deviation divide by the square root of n so you just substitute the values your sample mean we found that was 85 minus the population 80 divide by the sample standard deviation of 16 divide by the square root of 36 do the calculation and tell me what the answer is what is the answer 7.25 i have two answers 1.875 1.875 also getting one i also get 1.875 so that is correct so then it means the number a is correct so please check how you did calculate i didn't put it in the field i'm sorry all right i saw my way my mistake was thank you that's good um and then number two it says we need to find the p value now if our z-stat is 1.875 to go to the table we need only two decibels so we going to be using z of going to be using z of 1.88 because we need to round it off correctly it's going to be z of 88 now let's go back to the p value the p value says if it is less than we're going to go to the table value if it's greater than we're going to take to take one minus the table value if it's inside it we look at whether it's negative or positive and then otherwise so now let's go back to our question sorry i bet our question our z is 1.88 but our hypothesis therefore it means we shouldn't even have used the equal we should use the greater than so how do we find the p value so if i need to find the p value of this because the p value is the probability we've been doing this and the p value we go to the z table and we go to the positive side of the z table and we're looking for 1.88 so we're going to find 1.8 and 8 at the top and where they both meet that will be the table value it's not the p value it's the table value it's zero comma nine six nine nine go back and you say one minus zero comma nine six nine what is the p value 0.03 the p value would be one minus zero point nine six nine nine did i also get the right value yes it's nine six nine nine the p value is zero comma zero zero comma zero three zero one i don't know why they have a four there so the answer here is zero comma zero three zero one i'm gonna put the question mark there because i'm gonna assume that that is correct or not correct but let's or incorrect because one and and four might not be the same thing let me just double check the other statement for my sanity so the next statement states that at a at a five percent level of significance the null hypothesis is rejected so we can either use the p value and the level of significance so the rule says but that's the rule say the rule says if the p value is less than alpha we reject the null hypothesis so let's look at this what is our p value our p value is zero comma zero three zero one and our alpha they told us that it is five percent which means oh sorry i should not put the sign first must put my value zero comma zero five so looking at the two values i will say the p value is less than zero comma zero five therefore we're going to reject the null hypothesis that is the statement that we should be making c says at five percent level of significance the null hypothesis is not rejected and what we just said is going to be rejecting it so based on this it may see the incorrect statement and that's how you're going to validate and find the incorrect statement number g says at five percent level of significance the rule is to reject the null hypothesis if the test statistic is greater than or equals to zero comma uh it's equals to one comma six four five five now let's find that out this is based on the critical value because this is a critical value and the test statistic so we are basing it on this it says if our test statistic is greater than or equals to we do not reject or we reject if it's less than if it's if it's greater than we reject the null hypothesis so let's test that scenario so we know we need to find the critical value and we are doing a one-tail test so our z will be alpha and if i need to find z of zero comma zero five then i must go on to the table the other thing i need to go back to the summer to just give you the critical value versions that you need to always remember so these are the the values that we always always use so if we're going to use only z alpha then it means we're not dividing our alpha by two right we're using it as it is if we are dividing it by two that will give us the value that we are looking for so now going back so you can use on the summer table you can use the z alphas that you see on here so if i must mention something here because that's what i'm going to be using on the other side if you think about it in terms of the z values so the answer you get here of z alpha over two will mean something so in terms of the 80 percent which is zero comma eight zero your alpha value so we should have you should have a table with your alpha value your alpha value here will be zero comma two zero the alpha value here will be zero comma one zero on here it will be zero comma zero five on here it will be zero comma zero two on here is zero comma zero one but you also have your alpha i'm gonna put here your alpha over two values which means it's half of this value so this will be zero comma one which are the alpha values that we used here the probability values here will be zero comma one zero zero comma zero five zero comma zero two five and so on i'm just making that an example as an example now looking at our value here that we are looking for i can go to the z table and look for this z this value on the table i'm gonna make an example on the table and also using the z table so i'm going to tell you right now that we're going to use that value because i'm looking at z of zero comma zero five which is the same as alpha over two of zero comma one zero right which gives me the same answer so how do i find my z of zero comma zero five i will find it by going inside the table looking for zero comma zero five inside the table now this is one of those special cases where the difference between the two values will always be equal so zero comma zero five will be somewhere between those two values that we always have and that is one of the only special cases that way we use those two values where we use one comma six four five because the difference here is five and the difference on the other side is five and we end up having one comma four one comma six four five and that's how i know that this is correct the rule says if the the rule is to reject the null hypothesis if the test statistic is greater than one comma six four five because then your original rejection we did go and validate and find it it is one comma six four five if it falls here we reject the null hypothesis and that's how you will know that this statement is correct and that's how you will validate that and upper tail z test is applicable with its date that in the beginning there right there so i think with this one with the exception of this difference between four and one we can assume that the pv was correct at the beginning are there any questions are you still confused no questions no comments so we move on suppose the calculated test statistic for a one-sided tail test is minus 2.66 suppose further that the population standard deviation is null what is the p value for this test so here they are telling us that they calculated the z test and they found that that z test that they calculated is minus 2.66 and there is the other information that they have given us they say suppose this is for the lower tail test so what is the p value how do we find the p value and go back to the notes and remind ourselves if it's a lower upper and today how do we find the p value and what is the p value nobody knows how to find the p value we go to the table we go to the table which side of the table the negative the negative side we go to the negative side then we look for minus 2.66 and at the top six where they both meet and the answer is zero comma 0039 is it right if they would have said two take two sided two tail test if they would have said that how would we find the p value what will be our p value for our two tail it would have been two times zero comma zero three nine and that would have been equals to zero point zero zero three nine times two it would have been zero comma zero zero seven eight just add one and it will be zero comma zero zero seven seven eight and if it was a one tail it would have been zero comma zero zero three nine it do not change which one is which now how would I if for example they would have asked you to make a decision so for this one it's a one tail lower one tail lower so therefore it means the decision that you would have made would have been if your p value is less than that value you reject right so this would have been zero comma zero zero three nine the rule will remain if your p value is less than alpha you will reject your null hypothesis that will still stay but how will you find the hypothesis there how you find the p value will differ based on whether it is a two tail or a one tail ah sorry I made a mistake if it's one tail upper do you see where my mistake is at it's on this I supposed to say one minus it will be one minus zero comma zero zero three nine the only difference will be how you find the p value so one minus zero point zero zero three nine will be equals to zero comma nine nine six one zero comma nine six one and that's how you would find the p value in different ways so for a one tail lower the value on the table if it's a one tail upper you subtract from one if it's a two tail you multiply the table value by two that you need to always remember okay moving on to the next question consider a two tail or a two sided hypothesis with 10 percent level of significance which means your alpha of zero comma one zero and the degrees of freedom which then it is your degrees of freedom of 30 if the population standard deviation is unknown we already know based on the degrees of freedom it means our population standard deviation will be unknown what is the critical value for this test so we are doing a two sided test are we going to use t alpha and the degrees of freedom or are we using t alpha divided by two and the degrees of freedom using the first one or the second one we're doing a two tail are we using the first one or the second one which formula we're going to use to find the critical value without with previous three and the nodes yeah for a one tail we use for a two tail we use so my question is the same since it's a two sided test therefore it means it was a not equal which one are we using the first one or the second one you need to be able to identify these things because the minute you use the wrong formula to find the answer you're not going to get it right so we're using a two tail which means we divide alpha by two because there are two sides when it's one side we just use alpha when there are two sides we divide alpha by two to share the alpha value in the two areas so you are told your t alpha is zero comma one zero divided by two and you are told what your degrees of freedom is state so your t of zero comma zero five and thirty it means you need to go to that with table guys are you sleeping or maybe my mic is muted you're not talking to me let me see oh you are here you are here oh okay it's too hot which table are we using if i'm going to the t table right we're going to the t table the minute i put t there it means it needs to also indicate which table i'm using remember for when oh let's go back there before we go to the t table remember for when the population standard deviation is unknown and they give you the degrees of freedom like i explained there you should have these things in front of you and use them to check as i'm saying it or as i'm reading the question and say oh df degrees of freedom it means we're doing this we are on this column this is how you're going to identify things in the exam if you are not following me right now and following the things that i'm telling you and using them as your guide you will find it difficult when you are on your own because you need to to check your understanding as well so when i ask your question say it so that it makes you not to repeat the same mistake in the future right so talk to me let me charge my laptop the battery is almost finished okay so we know that we're going to use the t table right so we go to the t table critical values of t and we're looking for 0 comma 05 at the top and the degrees of freedom of theta 0 comma 05 and where they both meet that is the value we are looking for which is 1 comma 6973 1 comma 6973 and i can see yeah they only have two decimal places so therefore i can estimate it to 1 comma 70 and because it's too silent remember when it is too silent we have this area and we have this area this side is positive and the side is negative so it will be negative 1 comma 70 and positive 1 comma 70 and it is plus or minus 1 comma 70 because it's a two-sided if it was a one-sided you look at the sign the less than will be on the negative the greater than will be on the positive it's not always that but that's how you can assume the things will be happiness are we good are we good oh are we great yes we're okay i think more practice is what we need yeah and we get a very long question so various literacy group recommend a reading speed of 152 width per minute wpm for grade four learners a grade four teacher is convinced that the average reading speed for his class is more than the recommended reading speed in a sample of 36 grade four learners the average reading reading speed is 157 width per minute with a standard deviation of 22 width per minute okay now based on this information there are several things that we need to get right all we need to deduce from this so let's find out if you are able to identify things on this that can help us answer the question because the question is more about you are required to test the teacher's hypothesis that at five percent level of significance is great for class reading speed is more than recommended reading speed choose the correct statement from the list below and it looks like all this are more about the decision so there are certain things that we need to do in order for us to get to those decisions so the first one it says we reject the null hypothesis we cannot reject if we didn't do certain things if we didn't follow all four steps or five steps until we get to the conclusion which is the sixth step so it means we need to do all of them we reject the null hypothesis and conclude so it also is step number six so you cannot get to step number six without doing other steps we do not reject the null hypothesis we do not reject the null hypothesis there is not enough information all of them are about step number six and it means we need to do step one up until step five in order to get to step six so let's follow all the steps i'm going to ask a question you're going to respond back and we're going to do all this together not me all of us togetherness so let's start with step number one let's state the null hypothesis and the alternative hypothesis step one what is my null hypothesis and what is my alternative remember there's no right or wrong answer i'm going to tell you and i'm going to correct you and we're going to move on so try based on the information given here right so let's first start by identifying the things before we start with statement one and statement two to help us because those are the things that help us jog our memories in terms of what we need to be doing what is the 152 weight per minute that is your population mean um i'm gonna ask what does the more than mean is it the less than or is it the greater than it's the greater than it is the greater than um in a sample what is 36 36 is our n and what does this mean the average reading speed is and this is from the sample of 36 is this the population mean or the sample mean the 157 in a sample of 36 the average reading speed is 157 with a standard deviation of 22 is this the population or the sample sample mean it is the sample mean and with a standard deviation of 22 is this the population or the sample it is very important to recognize these things as early as possible is this the population or the sample is this standard deviation coming from the population or isn't coming from the sample i think from the sample it is from the sample because it's from the same sentence there is no breakage in this sentence it says the sample of 36 has the mean of 157 and the standard deviation of 22 so therefore this is also a sample standard deviation if it was not from the sample they would have put their population standard deviation or they would have mentioned it here at the beginning with the mean because i by mean giving you the first point of saying this is the population mean therefore the rest of the you cannot have two population means right okay so based on this information that we have plus the level of significance is your alpha based on this information then let's try and map out step number one up until step number six so i'm gonna say the population mean is equals to 152 because what i put in the null hypothesis has nothing to do with whatever the information they have given us here right really because we're not even going to use that it always has an equal sign the most important thing is what will go in your alternative what is that statement that will go into your alternatives what will that mean be the information is given in the statement already you have that you just need to tell me how do we say it the population mean is come on guys be alive it's greater than 152 because the null hypothesis and alternative we always use the population parameters right so what do we know so far already by doing this we know that we are doing a one tail one tail test right and it will be an upper an upper one tail test so it means there is only one region of rejection so step number two let's state what we are given we are told what the alpha value is it's zero comma zero five because it's five percent our n is 36 and our s because we're not giving the population we're giving the standard deviation is 22 what else we can find also the degrees of freedom or let's not start there let's not go there because I'm not sure what what makes me go to the study let's go to step number three what kind of a test are we going to be doing are we doing a z test or are we doing a t test look at your notes now we covered this at the beginning when we started if the population standard deviation is known we use the z test if it's unknown we use a t test now we looked at this is the population standard deviation known or unknown it's known the population standard deviation is unknown because we are given the samples standard deviation therefore what type of a t I'm already giving you answers what type of a test that we're using or are we doing 30 we're doing a t test we're doing a t test that's step number three step number four we need to find the critical value so we're doing a one tail test so our critical value will be t alpha and the degrees of freedom so our t of zero comma zero five and the degrees of freedom is n minus one what is our n n is 36 so it will be 36 minus one minus one then our t will be zero comma zero five and 35 so it means to go to the t table as we'll find our critical value on the t table where t is zero comma zero five probability and 35 degrees of freedom the t critical values of t zero comma zero five is our table and 35 our degrees of freedom and our critical value is one comma six eight nine six one comma six eight nine six four our critical value s one comma six eight i forgot now one comma six eight nine six that is step number four let's go to step number five we need to calculate the test statistic at number five we calculate the test statistic which is sample mean minus the population mean divided by the standard error which is which is the sample standard deviation divided by the square root of n and substitute into the formula we know that our sample what is your answer i'm getting 155 comma 848 55 yes 155 comma 848 no check your answer okay check that you are doing the right calculation it's 157 minus 152 divided by 22 divided by square root of 36 i get one comma 36 36 i also get that one comma 36 36 i'm gonna keep only two decimals there's no need for me to keep all of them so i get one comma 36 i'm gonna keep one comma 36 that is your t test statistic now we need to make a decision so making a decision it means we need to draw ourselves so step number six i'm just gonna write it here let's find another space i'm gonna write it here at the top step number six we make a decision so we draw the normal distribution because it is a greater than remember based on that therefore this is one comma three so what this is our critical value we use the critical value one comma six eight which is one comma six nine one comma six nine because i kept two decimals i'm just gonna keep one comma six nine that is the critical value which create the region of rejection anything this site going to reject the null hypothesis so now let's make that decision where does one comma 36 394 does it fall in the rejection area change my color to purple so this value where does it fall does it fall here or does it fall in the white area one comma 36 it falls in the white area in the white area in the do not reject area so therefore the decision is we do not reject the null hypothesis you can conclude that we do not reject the null hypothesis based on that information that we have now we can go and find the correct answer so the first two statements said we reject we reject so they are incorrect right so we left with only two statement statement number c or we actually left with three but i can also take away state statement number e because we have enough information because we were able to get all the information that we need to get to the conclusion to step six so c says we do not reject the null hypothesis and we conclude that the average brilliant speed is more than 152 d says we do not reject the null hypothesis and conclude that the average speed is not significantly different from 152 how do we conclude remember our null hypothesis stated that the reading speed the mean of the reading speed or the average reading speed is equals to 152 so it means is the same so which one of the two statements would you choose we will not choose number c because c says it is more than which reads the alternative hypothesis so based on that information we can also actually even eliminate that and the only statement would state that the mean difference is not or is not significantly different from 152 remember it can also be less than 152 as well so c is the only way you will conclude if you're not rejecting the null hypothesis then it means you're stating that the null hypothesis which says that they are not different they are the same it's what you are keeping if you are rejecting it therefore it means you are validating that the alternative hypothesis is correct and that is the statement that you will be you will be saying it's significantly different hey and that's how you answer questions one hypothesis you can see that it is very important to know all this information because if in the exam they only ask you about you making a decision it means you need to know how to find step number one about step number five to make that decision sometimes it also have to mean that you need to also include the p value in how you make that decision as well so you need to know the steps you need to know how to identify whether population standard deviation is given or not whether you are given the sample and whether you're doing the hypothesis testing for the proportion and this also goes to what we did the last time in terms of the confidence interval because there are three scenarios that these things can happen okay so let's look at most second last question and then we will be done with today's session in a sample of 200 the sample proportion is 0 comma 36 consider the following hypothesis the null hypothesis states that the population proportion is 0 comma 4 and this alternative states that the population proportion is less than 0 comma 4 what is the p value and they just need it right there what is the p in order for you to find the p value it means you need to find the z value first but before we do that we also need to identify several things on this question to help us because to find the p value we need to apply certain scenarios so what type of a test are we doing based on the alternative hypothesis are we doing a two-tail or a one-tail and I can hear my voice also like finishing depleting by the time it will be five o'clock my voice won't be there anymore because I'm the only one speaking in this session I can't rest my voice so are we doing a one-tail or a two-tail help me rest my voice I think it should be a one-tail we're doing a one-tail test and since we're doing a one-tail test it will also help us because for a one-tail test it will help us to understand how we're going to find the p value let's go back so that we are away on that we're doing a one-tail test and it's a lower tail test so the value we find on the table will be the value we are using and remember that we only use or rely on the z test and since we're doing for the proportion we're going to use the z test statistic right so those are the things that we need to be aware of so going back we are given the sample size of n we are given the sample proportion so yeah we don't even have to calculate the sample proportion if they didn't give us the sample proportion they've used x over n to calculate the sample proportion so they have given it to us we don't even have to calculate it the population proportion is always stated in the hypothesis testing that that will be less than and since we're looking for the p value I can put it in the bracket and say the sample proportion minus the population proportion divided by the square root which is the population proportion one minus population proportion divide by n that of less than because I'm following what the side is saying what is my sample proportion in 0 comma 36 minus my population proportion which is in the hypothesis 0 comma 4 divided by the square root of 0 comma 4 times one minus 0 comma 4 divided by n of 200 what this minus that value excuse me a little bit more I just want to understand sometimes there's a formula that the user the millimeter is I don't know which formula you are referring to tell me what is the formula okay so previously in some of the exercises or from the assignment or some way they might ask you to find the standard error but they gave you the sample proportion so you just use the sample proportion instead of the population proportion but for the real sampling distribution that's formula to calculate the test statistic we always use the same formula which is your sample proportion minus your population proportion divided by the standard error and your standard error in resistance since you have the population proportion will always be the population proportion one minus the population proportion divide by n and that is the formula that you will have to use and that is the formula that we use in sta if you check other books that might have used a different veil for example if you if you used books that are used by sta 1501 then they use a different formula in terms of the proportion but then they have the sample proportion as a copy and the population proportion as a normal p so you just need to make sure that you understand which one they are using and especially also when you go to the internet or you do you look at other people's youtube to try and understand some of the concept in your module this is what we use yes yes okay so have you found the answer what is your p value your sorry your z value my z value is minus 1.15 minus 1.15 do you also get the same yes the point is same okay what is the next step we need to go to the z table to go find the p value so we go to the negative side and look for 1.15 we go to the top and look for five and the p value will be 0.1251 0.1251 remember i'm not sure if you still have access to your my unicycle assignments so all these are the questions that comes from your assignment questions and remember this is one version so every one of you have different questions that they would have received in terms of question 12 so go and do practice activities based on your previous assignments as well you might have found that some of you have 120 there is the sample size the proportion might be 0.26 or 39 or something like that just go and see if you are able to do these exercises on your own based on your assignment question that you had at that point so these are for all the revisions that we have been doing so far up until now i have been based on your assignment that you did but one version of that assignment because every second student or third student would have received different questions or different data like the numbers might have been different to the one that we are using right now so use those assignment to practice okay so going to the last last last last question until we are done with the session and we are done with hypothesis testing and i hope at the end of this session at least there is some light in terms of how you answer questions related to hypothesis testing so the last question most of the school reported a decline in the number of absences following the educations department's lana transport program and school nutrition program in a sample of 200 schools from umzinyati district municipality 72 percent reported a decline in the number of lana's absentee the district manager is adamant that the true population proportion of the schools that reported a decline in the number of absences different in the number of absences is different from 78 percent formulate the null hypothesis and the alternative hypothesis and conduct a the hypothesis test for the true population proportion and a five percent level of significance we need to find which one of the following statement is incorrect that's what we need to find the incorrect statement so there are different things that we need to record 200 is our sample size 72 is our population proportion 78 is your sorry it's your 72 is your sample proportion and 72 is your population proportion because that's what they said they the true population proportion that reported a decline the absence is different from and the weight different means not equal right different means is not equal it is different okay so based on this we need to find what if the test statistic is correct the p value is correct the null hypothesis is rejected whether the alternative hypothesis is correct or incorrect and we need to make a decision so then we are back to square one we back to doing six steps of hypothesis so let's start the the first step of hypothesis first step your null hypothesis and your alternative hypothesis the population proportion is equals to zero comma seven eight the population proportion is not equals to zero comma seven eight because that's what they single ask from that set we are doing a two tail test step number two we state what we are given we know what alpha is our alpha is zero comma zero five the rest I've already highlighted them the population the sample proportion is zero comma seven two so our p if they didn't give us our p would have calculated it but they gave it to us so it's zero comma seven two step number three what kind of a test are we doing we doing proportions so it is obvious as a test step number four looking at this we are going to be using a p value so there is no need for us to go find the critical value so I'm just going to go ahead and calculate the z test that's that your sample p minus your population proportion divided by the square root of your population proportion one minus the population proportion divided by the square root of n let's substitute into the formula zero comma seven two minus zero comma seven eight divide by the square root of zero comma seven eight times one minus zero comma seven eight divide by n of two hundred if you get populations you just tell me how much seven two minus point seven eight divide by the square root fraction point seven eight times one minus point seven eight those bracket going down two hundred and I get minus two point zero five if I hand off correctly plus two point zero five that's step number four step number five we need to go find the p value we're doing a two table test a p value based on the information we had previously it's two times the table value because my z test is negative let's confirm that if we're doing a two tail test and your z value is negative you just take the z value is equals to two times the table value that's what I told you previously so I'm not lying to you right when I when I give you all this so it will be two times we need to go to the table value and find the table value from the z value of minus two point zero and zero five so go to the z table on the negative side we're looking for minus two point zero five so it means we're looking for two point zero here and zero five at the top where they both meet which is zero comma zero two zero two zero comma zero two zero two and that is equals to zero comma zero four zero four and step number six we make a decision and conclude oh because we're using the p value so the rule the rule says if the p value is less than alpha we reject the null hypothesis so let's conclude our p value is zero comma zero four zero four our alpha value is what they gave us what it was five percent right zero comma zero five zero comma zero four and zero comma zero five it's less therefore we reject the null hypothesis and state that the number of absences is significantly different from zero comma seven eight now I can answer my questions in the exam you don't have those negative I just want to show you all the steps and then we can come and answer the questions let's see if we can answer and find our incorrect statement the test statistic is two comma zero five we said it is negative zero comma two five let me change my color back to rate so we have our incorrect statement but let's validate our p value is zero comma zero four zero four that's correct the null hypothesis is rejected yes we are rejected the null hypothesis it's correct we can conclude that the proportion of school that reported the decline in the number of absence is significantly different from the number of absences is significantly different from zero therefore that statement is correct the alternative hypothesis is that the proportion is not equals to 78 the alternative that is correct so it means I've used that that and that as you can see that you have used all six steps of hypothesis testing to answer this question and that's why it is very very important to know all your steps of hypothesis and that on that note it is three minutes to five it concludes our session for today and that's how we conclude the session are there any questions any comments any query any anything you want to ask content related before I stop the recording and we can have general discussions is it possible whereby I can see uh the full video of the session at least I can see how do you come to more specialty that uh p gail whereby it's equals to two times the table I didn't understand that so I just want to get the full video I can see what's all about okay no problem any other question which is content related not administrative issues are we good are we happy with hypothesis testing I'm gonna stop the recording and we're gonna have a general discussions