 Good afternoon everyone and welcome to your session, I don't know, I think it's session territory also. Today we continue with the revision of the content. So we are going to look at assignment four questions, which was based on study unit eight and study unit nine. So studying with study unit eight, study unit eight was confidence intervals. So when we look at the confidence interval just to recap, there are two things you need to remember with confidence interval. We can calculate confidence interval for the mean or for the proportion. Taking a confidence interval for the mean, there are a couple of properties that you need to also remember. We need to remember if population standard deviation is no, what we need to do. And when it is unknown, what do we need to do, what do we need to do? What do we know as well, it got less of whether we do the confidence interval for the proportion or for the mean. We know that generally the confidence interval formula is given by the point estimate, which can either be for the mean, sample mean or sample proportion, plus or minus, the critical value times the standard error. That we need to always remember that that is the formula for calculating the confidence interval. But we also need to remember that only this two part happened when I was clicking, clicking. Okay. Anyway, so what we need to remember that the following, which now when I click, it does whatever it does. And let's not let the, what do you call it? Let's not let the technology determine what we need to be doing. Oh, come on. Serious. As my stuff, the PDF wanted to draw a line, it announced me to use the thing. Okay. So before the technology disrupted us, so we know this is the formula to calculate. The problem is I cannot write on the PDF. Can I just, can I just give me a sec to close it and open it again? Because now I cannot write. Okay. Now I cannot. So we're keeping. So we know we're doing confidence interval and for confidence interval, we have for the mean and we have the proportion. And we said for the mean, we always need to remember that when population standard deviation, if it's known what we need to do and population standard deviation, if it is unknown what we need to do. And I just explained that in terms of the confidence interval, the formula is point estimate plus or minus the critical value times the standard error. And that is the formula for calculating the confidence interval error. We also need to remember that the critical value times the standard error will give you what we call the margin of error or error of margin or margin of error. So the critical value times the standard deviation refers to the margin of error. Point estimate plus or minus the margin of error will give you the confidence interval. And with confidence interval, it has the lower boundary, which will be the minus gives you the lower boundary and the plus gives you the upper boundary. So you also need to remember how you find the boundaries. So when we calculate confidence interval for the mean, when the population standard deviation is known or given, then we use our point estimate, which will be our mean plus or minus for the lower boundary, upper boundary times the critical value. And since the population standard deviation is known, the critical value we're going to find using alpha divided by two. And the standard error and our standard error will be the population standard deviation divided by the square root of n. And that is your confidence interval for the mean when the population standard deviation is known. Critical value, we know that we're going to find it on the z table. That is where we're going to find the critical value. We need to divide alpha by z over two. To find the confidence interval for when the population standard deviation is unknown, we use the point estimate plus or minus for the boundary. The critical value will be t alpha divided by two and the degrees of freedom times the sample standard deviation divided by the square root of n. And that will give you the confidence interval for when the population standard deviation is unknown for the mean when the population standard deviation is unknown. Now, finding the critical value, we're going to use the t table. And on the t table, our degrees of freedom is n minus one. So it means the sample size minus one. And then we go find the critical value. For the proportion, we use p, which is the point estimate for the proportion plus or minus the critical value z alpha divided by two. For z, for the proportion, we always use z times the standard error, which is the square root of your sample proportion one minus the sample proportion divided by n. And that will give you the confidence interval. And we use the z table. If our sample proportion is not given, therefore it means they would give you the observation satisfying that and you divide by n to go find the sample proportion. Now, sometimes you also need to be able to find your value of your alpha by using your confidence level, your confidence level, which is one minus alpha. In order to find alpha, we need to use the confidence level. And you will need to know that for a 95 percent, the confidence level, 95 percent confidence level is 0.95 confidence level, which is your alpha of 0.05, because it's one minus 0.95, which will give you 0.05. You also need to know what your 99 percent confidence level is, which is 0.99, which our alpha will be, sorry, our alpha here is 0.01, that is our alpha. And you need to know how to take your alpha value divided by two and go find the critical value. So this is just to find the alpha, which is one minus 0.95 will give you alpha of 0.05, one minus 0.99 will give you your alpha value of 0.01 and other confidence level, like 80 percent, 90 percent, and so forth. You need to know how to use that. Okay, based on that information that I just shared, let's then look at the questions. What else am I missing? So you also need to know what else, nothing. I think we've covered more or less, but also some of the things that you need to also remember is if we increase the value of our standard error, what will happen to the, what happens to the confidence interval? Does it increase or does it become smaller? So remember that. Remember that if we increase the level of significance, where is your upper for this one and, sorry, lower boundary and upper boundary for that one. And if we do this one, what is your lower and your upper based on your level of confidence? If it's level of confidence, how does it affect, does it mean the lower confidence level is bigger than the smaller confidence interval? Or when your confidence level is bigger, the bigger the confidence level will be. So you just need to also remember all those things that we did. Okay, so now first question they're asking. A random sample size of 100 in a sample mean of 100, the population standard deviation is known to be 15. What is the margin of error of estimating a 99% confidence interval? The first question that you need to ask yourself is, am I doing confidence interval for the mean or the proportion? We're doing for the mean because they gave you the mean and the standard deviation. So it means we're doing the confidence interval for the mean. If you're doing the confidence interval for the mean, the next question is, is the population standard deviation known or unknown? They told you the population standard deviation is known or they even gave it to you. So sometimes they will not say is known, but they will just say population standard deviation is and you know that is known. So the population standard deviation is known. So it means we're using Z and the question is asking us to calculate the margin of error. Therefore, remember the margin of error is your critical value times your standard error. So it means we just need to calculate Z alpha divided by two and standard deviation divided by the square root of n. So what we need to be calculating is that alpha divide, why am I giving you the answers? I don't want to get to the answer before we calculate, divide by population standard deviation divided by the square root of n. So the first thing you need to do is go find your Z alpha divided by two. How do we go find Z alpha divided by two? Z alpha divided by two, we need to go first find alpha. We know that this is 99%. So a 99% confidence level is 0,99 which then it is alpha of 1 minus 0,99 which is equals to 0,01. So then you just substitute Z of 0,01 divided by 2 that is Z of 0,01 divided by 2 is 0,05. It's 0,005. Then it means we need to go and find this on the table. Go to the Z table. Standardize Z table and we're looking for 0,005. At the moment 0,004, 0,05 it should be that one. So it is two and that 2,58. So our critical value is 2,58. So we just substitute two. So what we're calculating is 2,58 times our standard deviation is 15 and our N, that's our N, it's 100. And what is the answer? 3,87 which is E. There are two correct answers. I've got a query regarding this first question. Since we were given two same answers, but when we chose one they gave us wrong. So I have a query regarding this. I think they need to do a remark on this question. I guess probably you will get a format. Everybody will get a format for this question. They gave us zero. No, when they recalculate, remember when you look at the score. So there's two different things that you will see your answers or what do you call that? Your results. You will see your score which will give you how much mark was this two points. You will see on your assignment, on the online assignment, you will see this two points or zero. But when you go to the grade book, because grade book is 100%, you will see your percentage. And for questions like this, part one, they would have given you formats, whether you got it right or wrong, because of the errors that exist on those questions. So your grade mark will be higher than your question, like your assignment score, because this is the score points that you will receive. You will see on the side. But when you go to the grade point, it will be different. All right. But the answer is 3,87. The estimate below a 95% confidence estimate interval estimates for the population mean calculated using the sample size of 20, 40, and 60. Which estimate must have been calculated using the sample size of 20? That was a very tricky question. What are they asking you to calculate? Confidence interval for the mean known. Yeah. But they say which estimate must have been calculated using the sample size of 20? So the challenge with this, unless maybe, I don't know, if this was a full question, as asked like this, or there was additional information somewhere here given. Or if it relates to the part that we calculated above, where we were given the 100, sample mean, and the standard deviation of 15. Okay. So let's assume we're using that one where we were given. Oh, sorry. The sample mean of 100. And what else were we given? The standard deviation of 15. Which is the previous question that I am looking at. So if now we need to calculate a 95% confidence interval, so we need our 0,95, which means our alpha will be 1 minus 0, 1 minus 0,05. So our alpha will be 0,05. 0,05. And if our alpha is 0,05, therefore our alpha divided by 2 is 1,96. We know that because 0,05 divided by 2, 0,05 divided by 2 gives us 0,0250, which then is 1,96. So if our z value is 1,96, so I'm going to calculate our mean plus or minus our critical value, which is z alpha divided by 2 times the standard error, which is the standard deviation divided by the square root of n. Our mean is 100 plus or minus our critical value, which is 1,96 times our standard deviation is 15 divided by, now they say must use the sample of 20 divided by the square root of 20. Calculate that. So then it means you are going to split this into two at the end. Let's see. Is there only three answers to this question? Yes, there are only three answers to this question. It is 100 minus, oh sorry, that's 1,96. 1,96 times 15 divided by the square root of 20. And that does not even give me, it gives me 93 on the minus 93 comma 42. What do you get? Yeah, same. And then on the upper one it's 106.5.6. And then on the other side it's 106. Okay, let me do that. 106.57. Okay, change your sample size from 20 to 40 because I don't know what they are trying to achieve with this question. So if I change the sample size from 20 to 40 on the minus side I get 95. Let me put it there. I get 95.35. And on the positive side I get 104.65. And then if I change it to 60. If I change it to 60 then the upper one would be 103.79. It becomes 103. That's the upper, not the lower. Oh, the upper one. And the lower would be 96.2. 96.2 and 103.2. I'm gonna keep it to 80. Okay, so if we have multiple values here, I'm not sure. So if I'm assuming, if we use the 101 and then here I'm going to assume that they're asking you to check whether, if you have a sample size of 20, 40 or 60, whether your confidence interval are going to be bigger or smaller or, and that is why I did it this way as well. So when it is 20 we know that it is 103 and 101. When it is 40 is 95 and 104 which is bigger. And then when it is 60 is 96 and 103. You guys see? Yes. Yes, I remember this question from from the Simon. I also struggled, but when I read the question, what immediately came to mind, remember there's something you told us about the rule when it's smaller, it's wider and when it's bigger it's narrower. Yes. That is what I applied to this question and I got this one right by applying that method. Yes, that's what I was trying to do with this as well to say when it's big, but yeah because we're talking about the sample size. So let's let's look at this one. So for a 20 is if I had to draw this on a number line like that. So 20 is 93 and 101. That is 20. So this is 20. 40 is 95 and 104. That is, that is 40. And then 60. 60 we got 96 which starts somewhere here and 103. Why 60 gets less than 40? Something's not right. I'm sure for 40 we got 104. On the plus side, yes it was 104. It's very weird. Okay, so if I look at these values then 20 should be less. 78 is less. Oh, sorry. This should be 106. Yes, sorry. My bad. This should be 106. For 20 should be 106, not 101. It's 106. So 20 should be wider. So it should be less and wider. So 20 will be bigger and 40 will be 95. 104 and then 60 will be, that is 60. So if we look at all of them, so therefore it means the correct answer should be C because we're looking at 20 and 20 is wider. All right. 20 is bigger. Then 40 will be in the middle and then 60 will be there. So that will be the wider one because 78, 78 and 121 will be bigger. 84 will be in between and 112 will be there. So therefore this bigger one, the outer one would have been the 20. This one is the 40 and this one is 60. So this is 60, 40 and 20. So the answer is C. So you just need to know the rule on how to apply it because on this one, since they didn't give you additional information, how would you know is just make up some dummy values because I realized that they used the hundreds and that's why I used the hundred and the standard deviation of 15, but they could have used any other value. That's why he's not going to match 100%. But at least it gives us an idea in terms of when the sample size is smaller, the confidence interval is bigger. So when n is smaller, so that you need to remember. When n is smaller, the confidence interval becomes bigger. When n is big and then the opposite, when n is big, which is 60, the confidence interval will be smaller. So that is what you need to remember. So n is smaller, the confidence interval is big, n is big, the confidence interval is smaller. Answer is C. The Department of Basic Education alias finding from a survey of 150 lenders suggests an average travel time of 114 minutes from home at one of the schools. Assume the population standard deviation of lenders travel is 72, construct a 95% confidence. So we are told what n is. We need to buy that mate for here so that we can sit here on the floor and play games. And our mean also need to buy that thing, so that we can put it there. And the sample mean is 114. And what else are we given? Assume that the population standard deviation, so we are given the population standard deviation of 72. So it is known. It's very important to identify that construct a 95% confidence interval. So it means we know that we need z alpha divided by two. What is z of 95% percent? This one you should know it by heart. This one you don't even have to go and look it up. You should know it. Okay so calculate or find your confidence interval estimate plus or minus z alpha divided by two and the square root of n. I will check your answers on the chat to see how much did you get? I don't have access to the chat but my answer is b. Okay. How did you substitute? What is your x bar? 114 plus or minus your critical value of 1,960. The population standard deviation 72 divided by the square root of 150. And on the minus side? Minus side 102,48. 125,52. If we didn't round it off, we can round it off from there it was 7776 and on the plus side what do you get? 125,522. Which will be if we round it off to two decimal b. A household survey based on a sample of 50 households in a rural area or rural village found the average household income to be 3.5. With the standard deviation of 8,000 construct a 99%. Have they given you population standard deviation? Yes. How do you know that they gave you the population standard deviation? They say from a sample of 50 households they found the mean to be 3,500 the standard deviation to be a thousand. Have they given you the population standard deviation? The answer should just be no because they didn't give you the population standard deviation. They gave you the standard deviation from the sample because they say from a sample of 50 they found the mean to be 3.5 which is your x bar of 3,500 and the standard deviation which is s of 8,000. So the population standard deviation here is unknown. Sigma is unknown. You need to be able to read the question to make sense of it whether is this a population or a sample standard deviation. Construct a 99% confidence interval so we know that our n is 50 and our alpha will be 0,01 so we need to go find the critical value from the t distribution table which is t alpha divided by 2 and the degrees of freedom and we know that the degrees of freedom is n minus 1 which is 50 minus 1 which is equals to 49. So here we'll have t of 0,01 divided by 2 and 49. We need to go to the t table to go 5, t of 0,005 and 49. Going to the t table to 4, 0,005 is the last column, t of 49 degrees of freedom of 49 is 2,680. Our mean is 3,500 plus or minus our critical value 2,680 that's what we got times standard deviation of 1,000 divided by the square root of n. For the minus it's 3,120,9907. 3,120,9907. 0,765 and some numbers. And from the plus side, 0,009,923 looking at the answer plus other numbers so don't forget those other numbers there. Let's see 3,120,99 and 3 if we round it off 3,799 we round it off this will be 1 so 3,879,01 which is option number 1. Consider a t distribution with the degrees of freedom of 27. Which one of the following statement is incorrect? So the degrees of freedom is 27. Just want to go slow so that I don't give you the answers. Okay so what do we need to do now? They say a says so we need to find which statement is incorrect so we know what the degrees of freedom is. Number a the t value associated with the lower tail probability of 0,01 is that B 95 percent confidence or the 95 percent of the area under the t distribution falls between that and that the value t the t value associated with the lower tail probability of 0,025 will be that a 90 percent of the area under the care falls between that and that a 80 percent falls between that and that because these are just critical values all of them whether they chose whichever one is the critical values so let's see which one is the incorrect one we know what the degrees of freedom is let's go find our t of 0,01 and 27 t of 0,01 which is that column so we can come to this one which is the second column and 27 because that is our degrees of freedom we find that it is 2,473 let's go to our question it says the lower tail where is the lower tail the lower tail will be this side for the critical value so this side we know it is a minus so it will be 2,473 on their side it would have been positive if it was in the upper side so that is one that is what we wanted that is that which in turn it says a 99 percent confidence interval on the lower side it would be minus 2.3 that is correct now I want to skip or let's not skip it anyway because I wanted to skip this one's where it has the two values and only use the one where there is only one value so here let's use see let's look at see the t value associated with the lower tail while we still at the same point so we're still also looking for the lower tail let's go look for t of 0,025 if you go find it so now 0,025 and the degrees of freedom so you need to go to the table t of 0,025 is the next dough she's that one is 2,052 2,052 and they say in the lower tail this is the lower tail which is 2, will be minus 2,052 so it is a minus 2,052 if they would have said in the upper we know that for any distribution yeah it's zero so the side will be positive and the side the left side will be negative the right side will be positive so that is correct so now let's move to the one where they've got the two values but they also give us a 95 percent confidence level so at 95 percent they say if 95 percent of the area and that the distribution is between that and that what they're asking because with the other one we were looking at only the small portions side so what they're saying is can you find out if let's use the same the same diagram that I have here they say if if we have this critical value what does this area represent is it a 95 percent confidence so we'll have the minus critical value there and the positive critical value does it give you a 95 percent so what we need to do is we need to go find our alpha alpha for 95 is 0 comma 05 so alpha divided by 2 is we did calculate it what did we find 025 we found that it was 0 comma 025 so did we find what is the critical value if we use this to go find the critical value is minus 2 comma 052 on this side and 2 comma 052 on the other side so therefore this is the incorrect one just to prove to you because don't take my wait for it and say oh but it's because you just want to give us that answer let's look at number D and number E and see if that area is the same so I'm just gonna remove all these values that we have the concept still works the same so what we need at 90 percent what is alpha alpha at 90 percent is 0 comma 1 0 comma 1 alpha divided by 2 it's 0 comma 05 it's 0 comma 05 so now we need to take our alpha of 0 comma 05 and 27 and go find the the the critical value 0 comma 05 and 27 where is 0 comma 05 1 comma 703 1 comma 703 so this side will be minus 1 comma 703 and this side will be positive 1 comma 703 do we have that on the answer yes definitely we do let's look at 80 percent you do the same at 80 percent 80 percent is 0.20 so what is 0 comma 2 0 divided by 2 0 comma 1 0 0 comma 1 0 so now take 0 comma 1 0 and go find the probability so where is 0 comma 1 0 is the first column which is 1 comma 3 1 4 so therefore it means for that one it's minus 1 comma 3 1 4 and this side it will be positive 1 comma 3 1 4 is it the same yes it is and that's how you will find the correct answer of which it's the incorrect answer yes okay a simple random sample of 46 from a normal population results in a sample mean of 300 and a sample standard deviation of 50 construct a 90 percent confidence interval for the population mean choose the correct answer so we are given our n of 46 we are given the sample mean of 300 what else are we given population or sample standard deviation sample standard deviation and on this one it's easy to recognize because they did say a sample standard deviation on the other one they didn't say but you need to be able to read the question in order to identify whether this is the sample or the population the sample standard deviation of 50 and they are asking you to construct a 90 percent confidence interval which means your alpha will be equals to 0 comma 1 0 so you need to go find z alpha divided by 2 which is z of 0 comma 1 0 divided by 2 z of 0 comma 0 5 and that is is the 1 and 1 optional z value that we use with three decimals so 0 comma 0 0 comma what are we looking for sorry my bad 0 comma 0 5 0 comma 0 5 which is 1 comma it is the only one way it uses two values or two yeah three values 1 comma 6 4 5 for 90 percent you need to know that remember there is some way on the notes when you go write the exam you must have some key things next to you that will assist you with ease of answering some of the questions like the z the critical values z alphas you need to have a table that i think we did discuss this at some point z of alpha because even when you answer questions on hypothesis testing you need to know the z only for the z it works only for the z for the t distribution it will not work because t distribution depends on your level your degrees of freedom but for the z you can create and say for 90 confidence interval this will be 1 comma 6 4 5 and you can say for a 95 this is 1 comma 96 but you also need to know for the alpha when it is not divided by 2 what will it be okay but that is for another day this discussion so now construct a confidence interval so and you should have told me that this is very wrong can i ask something for the sample if a sample standard deviation is given isn't it unknown and we're supposed to use the t t table because that is what i was saying yes and that's why i'm saying you should be picking up this that this is wrong you see that's why i'm deleting everything that we just did now because you need to look at the samples standard deviation if it is unknown we use a t test we don't use a z we use t alpha divide by 2 and the degrees of freedom because the population standard deviation is unknown so our alpha still stays t of 0 comma 0 5 and the degrees of freedom will be our n is 46 so it will be 45 so you need to go and find your t alpha divide by 2 and the degrees of freedom and s divide by the square root of n you need to go find the critical value on the table on the t table you're looking for 45 which is the last one and it's 1 comma 6 7 9 and 0 comma 0 5 1 comma 6 7 9 your mean is 300 plus or minus 50 divide by the square root of n which is 46 for the minus it's 287 comma 6 22 287 comma 6 22 is 312.37775 37 there's a 3 comma 1 2 3 yeah triple 75 okay so the answer is in two decimal we can just round it off to two decimal so that will be two and that will be eight very eight so let's see 287 287 so it's number c consider the following application exercise from the previous assignment most of the school reported a decline in the number of absences following the education department's learner transport program and school nutrition program in a sample of 150 schools from Joel Gaby i don't know if i'm pronouncing it right district municipality 114 schools reported the decline in the number of learner absences construct a 90 confidence interval estimate of a population proportion of schools that they protect that a decline in the number of learners and chose and choose the correct answer from the list of options what confidence interval are we looking at here for proportions and if it's proportions then we're going to be doing the sample proportion plus or minus the critical value times the standard error which is sample proportion 1 minus sample proportion divided by n so what are we giving n of 150 we're giving x of 114 and we can find p because p is x over n which is 114 over 150 what is that 0.76 0.76 and we need to find the critical value of 90 percent which is z of 0.10 divided by 2 which is z of 0.05 and we did discuss this which is 1.645 substitute into the formula our p 0.76 plus or minus our critical value of 1.645 times the standard error which is 0.76 times 1 minus 0.76 divided by our n of 150 minus 0.7026 on the upper or lower on the lower side on the lower 0.703 when I round it off let's keep it on decimals it's 0.7026 okay yeah I'll just keep some of them and then on this on the plus side it's 0.81736 we can then round it off to three decimals when we look at the options okay 0.7 so that is not that is not that is not 0.7 but it's 0.73 because when we round off that it's going to be three and when we round off seven is going to be seven so the answer is option one others are you following yes or are you lost you must talk so that then when we leave the session you're not confused or what is happening okay so I think that concludes chapter eight okay so now we move into the hypothesis testing with hypothesis testing with hypothesis testing also you need to remember whether it's hypothesis testing for the mean or for the proportion and if it's for the mean also is the population standard deviation known or is the population standard deviation unknown unknown you need to also remember all that because when it is known we use z when it's unknown we use t when is the proportion we always use the z the other thing that you also need to remember are the hypothesis testing steps so you need to remember to state your null hypothesis and your alternative hypothesis I'm going to be like that so we know that with a null hypothesis there is always an equality sign whether it's less than or equal or greater than or equal but there is always an equality sign with the alternative which is your most important hypothesis testing statement there is no inequality so it will say either less than or or it's going to say greater than or it will say it's not equal and both of them whether it's less than or greater than or whether it's not both all of them they give you a sense in terms of the type of a test you're doing so this will be a one tail a one tail test and this is a two tail test it's very important to know the difference because we also need to make a decision at the end and that is very important step number two you need to be able to state what you are given in relation to n in relation to alpha and in relation to other values then step number three you need to be able to state the test that you are doing whether you're doing a t test or as a test based on the information given from previous that number four you need to be able to find the critical value critical value and based also in terms of whether you're using a z or a t how do you find your critical value and also based on your now your alternative hypothesis whether you're going to find your z alpha or alpha over two or your t alpha or alpha over two for a two take for a one tail we just use z for a two tail we divide the z by two step number five you need to be able to calculate calculate the test statistic whether it's a z stat or it's a t test step number six you will need to be able to make a decision and whether you're making a decision based on a two tail test or making a decision based on a one tail a one tail test so this one will be if your hypothesis testing your now your alternative hypothesis says it's less and this is when it says it's greater than when it's greater than and this is when it says it's not so you need to be able to make use of that the same way as this will the t for that will be for the not equal and equal and this one is when you do not equal so you need to know all those steps of hypothesis testing what else do you need to know making a decision especially for the z if only for the z either for the proportion of the t test you can make there are two scenarios scenario one is what i explained there using the critical value and the z state or the the critical value and the z state is the the one that we used when we were explaining the steps scenario two is using a p value and your level of significance and alpha p value and alpha the decision on this one says if the p value is less than alpha we reject the null hypothesis that is the rule whereas with the critical value it says once you have defined your critical value so if it's a two-tail test once you have defined your region of rejection whether on the upper and the lower you make a decision if it falls in the rejection area you reject if it falls in the rejection area you reject otherwise you do not reject with the p value the decision says if the p value is less than your alpha you reject how do you find the p value also depending on whether you're doing a two-tail test or a one-tail test so finding the p value if it's a two-tail test the value you find on the table you go into multiply by two but only if the value is on the negative side if your z value is negative if your z value is negative the value you find on the table will be two times the value you find on the table if your p value will be and let's put it this way let's put it this way let's rewrite it this way so that you can understand so if your z value is equals to negative then the p value will be equals to two times the table the table value that is for a two-tail test when your alternative says not equal if your z value is positive then your p value will be two times one minus the table value if it's for greater than or equal the same thing will happen so for a less than so if your z uh your alternative sorry i must use the alternative if your alternative was greater than then you say the p value is equals to one minus oh sorry equals to one minus the table value or something like that when it's less than it's equals to the table value so your p value will be equals to the table value for the less than for the greater than one minus the value you find on the table because on the table it will be showing you the positive one will show you the value the greater than value as well for when z is positive when z is negative then the p value is the value you see on the table okay so i don't think we will finish everything in one day we write k rule on on this on on wednesday and then we'll do the assignment two as well so we are on number eight and i think they are dating questions so we might not finish all of them in a sample of 36 the sample mean is 83 it is also known that the population standard deviation is 16 you are required to use the information to test the following hypothesis given that the null hypothesis is equals to 80 and the alternative is greater than eight so we're doing a one tail test so it means there is one origin of rejection the critical value we're going to find it by using alpha and so forth they have given us the population standard deviation so it means we're going to do the z hypothesis test which one of the following statement is incorrect so we're looking for the incorrect statement the test statistic is one comma two eight six so it means with the test statistic they want you to calculate z is equals to because this is that stat is equals to the sample mean minus the population mean which we will find on the hypothesis testing divided by the population standard deviation divided by the square root of n which is the standard error so substitute the values onto the formula our sample mean is 83 our population mean is given in the hypothesis is 80 divide by standard deviation 16 divide by the square root of n which is 36 what do you get one comma one two five and therefore it means we already have our incorrect statement anyway let's assume that that was not the incorrect one the p value we need to take one comma two three so we need to go take z of one comma one two three because we need to round it up to two decimal and that is to the one comma three which is zero comma eight seven which our p value will be given by one remember I said p value will be given by one minus the table value one minus zero comma eight I forgot now zero comma eight seven zero eight eight seven zero eight do you get the same answer might be that I'm using the wrong table than the one that they used but that should give you one minus point eight seven zero eight it doesn't give me that I have one comma what what what what what how did they get that because this is one comma one even if I say one comma one two I still will not get the same answer as what they have there so then it means we have two answers that are incorrect one comma one just double check something here I think that these are typing air on the answer or something I don't know but if the answer if the z value is what did they get they get a and a is the right one so the p value one comma three why do they have one comma three one minus eight seven it should be one comma two nine two what do you get even if I use the other ones I'm not going to get that option that they gave I think zero comma one two nine two is the truth is to the answer V yeah but I think they use that they they went to the table and chose the wrong like instead of choosing eight seven zero eight they chose the if you say one minus point eight nine zero seven oh but also it will not even give you what they have there because I'm trying to think how did they even get there to that value because you won't even find the opposite one three on the opposite if you go to the negative three three oh oh three three oh oh six oh seven those are the only three three oh values you will get which are not even but the one that we have we get the answer for it so that's what I'm wondering how I think they I don't know oh I'm looking at the wrong one one three so their answer is 13 oh three 13 do you see any 13 oh three here because these are the dead things that there is no 13 oh three unless on the five decimal table or some way since we have multiple tables oh gosh let's see I see they have the tables this one it's not this one to get the other one which has the constantly used this one doesn't have I want the table with five decimals will I ever find one like that let's see this one has four decimals as well or the one with three decimals maybe okay anyway I'm not gonna find an answer for for that one because then then means this one also is not right at 10 percent why do I have one come eight seven oh eight which is zero comma one two nine two I'm gonna assume that they rounded off some way that's how they got because if I rounded off to two decimal will get the same answer but anyway at 10 percent level of significance the rule is to reject the null hypothesis so now we can go and find at 10 percent level of significance means because we're doing a one tail we need to go find z of zero comma zero comma one zero z of 10 percent so 10 percent is one comma one zero one comma one zero which is one comma two eight one comma two eight so if that is the case our critical value is one comma two eight then it means we need to go and see if we can reject the null hypothesis and this because this is one comma two eight that's the decision we're doing a greater than so the original rejection is on the positive side and it is one comma two eight we take our z test and we allocate where it is it is in there wait wait wait wait wait wait wait so here they say at 10 percent significance level the rule is to reject the null hypothesis if the test statistic is greater than or equals to one comma eight two eight five so we're not using the one two five we must use the test statistic that they gave us which is one comma two eight five so where does it fall no man that is our critical value what is our critical value why do I have 120 128 because our critical value was 128 that's what we found the z value of one two eight falls in the rejection area so therefore we're going to reject the null hypothesis because it falls in the rejection area so question number c is also correct so that is correct we reject the null hypothesis if the test statistic is equals to a greater than 128 because we did find our critical value to be one comma two eight at 10 percent level of significance the null hypothesis is not rejected if we base the answer on our correct test statistic it will fall in the do not reject area therefore it also correct and this is a an upper tail area z test which is also correct the only correct that is the only answer that is not correct is option a i'm still worried about the small errors that are cleaning up on most of the questions okay so suppose the calculated test statistic is a one sided tail test and the test statistic we found that it's minus two comma seven four suppose further that the population standard deviation is known so if that is the case then this was a z test and it was a z test of less than because we were doing the hypothesis of a lower tail minus two point seven four or what they want you to do is go find the p value so go to the z table and find the p value of minus two point seven four so go to minus let's move all the ink minus two point we're looking for minus two point seven four minus two point seven four so two point seven and a four zero comma zero zero three one zero comma zero zero three one happiness yes consider a two sided test with five percent level of significance and a degree of freedom so now yeah they say a two sided test so it means it's not equal with a five percent level of significance which is our alpha and the degrees of freedom so our alpha of zero comma zero five and the degrees of freedom of 25 if the population standard deviation is unknown therefore it means we're going to be looking for the t critical value which will be alpha divided by two because it's a two sided test you need to be very careful as well read the question carefully alpha divided by two and the degrees of freedom our alpha divided by two which is zero comma zero five divided by two and the degrees of freedom of 25 which is t of zero comma zero two five and 25 so we need to go to the t table and look for zero comma zero two five zero comma zero two five and the degrees of freedom of 25 and your critical value is your critical value is two comma zero six zero so your critical value should be number one two comma am i getting it right two comma zero six it's two comma zero six happiness so you need to read the especially for these two questions so because this one said a one sided test and this one says a two sided test so you need to know that for a z for a not for a z for a one sided test we not dividing or multiplying the p value or dividing or anything like that and we use the z test because they told us that the population standard deviation is known on this side because they say the population standard deviation is unknown and we're doing a two tail test we need to be able when we find the critical value to divide alpha by two and use the t table based on the information given okay number 11 various literacy groups recommended a reading speed of 82 words per minute a grade one teacher is convinced that the average reading speed for his class is less than the recommended speed in a sample of 32 hour n the average reading which is the mean is 77 and the standard deviation is 15 so you must be very careful in a sample we've got a mean of 77 and the standard deviation with the standard deviation of 15 so what are we given we are given the sample standard deviation because it's from the same sentence the standard deviation that is given there is your sample standard deviation you are required to test the hypothesis at five percent level of significance so now what they gave you here are the rejection areas or the rejection regions so you should have already done all step one up until step five so to do that it means you first need to calculate your step let's find the critical value first do critical value of alpha remember less than that's the first other thing that you need to also remember less than is one sided so you're doing a one tail a one tail test so if you want to take shortcuts and not do the hypothesis test and all that you need to be able to identify key things that are given to you in the question so we're doing a one tail test alpha and the degrees of freedom they told you what alpha is is zero comma t zero comma zero five and the degrees of freedom they told you that they are 32 so the degrees of freedom will be 32 minus one so you go find the critical value on the table 31 and zero comma zero five 31 and zero comma zero five and 31 31 it's one comma six nine six one comma six nine six that is your critical now we know what the region of rejection is at because it says it's less than so this side minus one comma six nine six that's where the region of rejection is at and that is not the end we need to go find the test statistic the z state by calculating the sample mean minus the population mean divide by the sample standard deviation divide by the square root of n which our sample mean is 77 minus our population mean is always given in the question but now because we don't have the standard the hypothesis testing it's 82 it was given in the statement it's 82 divide by the standard error of 15 divide by the square root of 32 what is your z did you calculate it what do you get minus one comma eight eight five six eight let's keep it to two decimals eight eight so minus one comma eight nine yes to two decimal so you must also remember on the number line if this is zero this is one this is two so a bigger number on the negative side refers to a smaller smaller vein so since our critical value is one comma six nine our test statistics is one comma six so it falls in the rejection area net our test statistics falls in the rejection area because our test statistics is one comma eight when our critical value is one comma six so it will fall in the rejection area so now it means we're rejecting the now hypothesis so now let's look at all the statements and see which one is choose the correct statement so we're choosing the correct statement a says we reject the null hypothesis and conclude that the grade one reading speed is less than 82 remember your null hypothesis would have been the mean is equals to 82 the alternative would have said the mean is equals to it's less than 82 so we're saying we're rejecting this statement therefore we're saying this one is true so we reject the null hypothesis and conclude that the grade one average speed is less than 82 widths per minute that is the correct way or the correct one so but let's read the others we do not reject we know that we are rejecting so that one won't be correct there is not enough information provided to conclude but there is enough information because they have given us everything we need to do the hypothesis testing we reject the null hypothesis and conclude that the grade one reading speed is not significantly different but we know that we are rejecting and we know that it is different because we are accepting our alternative of less than so the second part of that statement makes it not correct because we know that there is a less than a recommended speed we do not reject that statement is incorrect because we are rejecting the null hypothesis the only statement here is a it's a in the sample of 100 the sample proportion so now we're looking at the hypothesis of proportion the sample proportion is 32 consider the following hypothesis calculate the p value so in order for you to calculate the p value you should calculate the z value so the z is p minus or your z step is p minus the population proportion divided by the population proportion one minus the population proportion divided by n so we have 0.32 minus 0.4 divided by the square root of 0.4 times 1 minus 0.4 divided by 100 and what do you get minus 1.63 minus 1.63 now the other thing you need to also go back and look at is the hypothesis testing here there is a typing error this should be an alternative hypothesis so the hypothesis test says we're doing a one-tail test so it means the value we find on the table if we're doing also a one-tail on the lower boundary which is in the negative side and we know that the table the z table contains the legs the probability of a less than so we just go to the z table and look for minus 1.63 z minus 1.6 1.63 which is 0 comma 0 5 1 6 0 comma 0 and 5 6 and that's how you will find the p value oh I thought we were not going to finish we are done so this one has more answers so we need to scroll to the bottom most of the school reported a decline in the number of absences in school program a sample of 200 schools from amatole district 85 percent reported a decline in the number of absent learners the manager is adamant that the true population of the school was reported in the number of absent is different from 78 previously formulate a null hypothesis and an alternative hypothesis and conduct the test for true population at five percent level of significance okay so state statement number one state null hypothesis and an alternative hypothesis so based on the information we need to also be very careful about the the two values so the amatole district is your sample because it's one of the district from from the the the department of education districts it's only one out of many so the population proportion which they told you there they say it is 78 so you need to know that for all the hypothesis testing we always use the population proportion which your null hypothesis will state that the population proportion is equals to 78 percent and the alternative will state what they are saying they say it shows that there is a reported a decline in the number but that doesn't mean that it's different because they say they say you need to test the hypothesis but they didn't tell you that the decline is in relation they know that there is a decline but they are not saying that that is less than or some is less than the municipality absenteeism is less than or no the amatole district is less than the population one so they're not saying that so it means here we're talking about the not equal situation so this will be the probability that it is not equals to 78 percent which will mean that we're doing a two-tailed test the number two we need to find the critical value let me just double check if all that so let's not find the critical value let's calculate the z test so that we can calculate answer the question which is p minus the proportion divided by the square root of population proportion minus the population proportion divided by n and the sample proportion is 85 0.85 minus 0.78 divided by the 0.78 times 1 minus 0.78 divided by our n is 200 calculate that so that then we can go and find the p-value as well and then we can answer the question two comma three nine two comma three nine so because we're doing a two-tailed and the answer is positive we need to go to the positive table two comma three nine we go find two comma three and nine is the last column which is that so we need to say one minus this value of zero comma nine nine one six what do you get zero zero zero eight four so we find zero point zero zero eight four multiply that by two zero comma zero one six eight two comma zero comma zero zero eight four i just want to write it here zero comma zero zero eight four and you find the p-value is zero comma zero one six eight that is our p-value and we can make a decision based on the p-value or the critical value it doesn't really matter which one you use we know the decision says if the p-value is less than the critical value since we didn't go and find the critical value we can use the p-value if the p-value is less than alpha we reject h h not so now our p-value is zero comma zero one six eight and our alpha they did give us the alpha they said it's at five percent level of significance so our alpha is zero comma zero five so our p-value is less therefore we reject the null hypothesis so now let's answer the question the alternative hypothesis is that the the alternative state that the population proportion is not equals to 78 that's what we stated not equal and we can conclude that the the proportion of school that reported the decline of absence is significantly different from zero comma zero eight we can conclude that the population proportion that reported the decline on the number is significantly different so because we're rejecting the null hypothesis therefore we're saying if they are different because we're saying they are not equal because if they were equal we wouldn't be declining the or rejecting the null hypothesis so therefore this statement is also correct so both of them are correct the null hypothesis is rejected we rejected the null hypothesis based on the statement that we made that's true then the other two the test statistic is two comma 39 did you round off correctly because they it says it's two comma 38 what did you get yes the answer was two comma three eight nine so when we round it off it's two comma three nine okay so two comma three eight nine so you need to make sure that you round off correctly so that is correct as well the p-value is zero comma eight four the p-value is zero comma zero one six eight is not zero comma zero eight four so that is the incorrect one so because of a two-tail test that's why the p-value will be because the p-value let's go back to when we were discussing this when we find the p-value for a not equal we multiply the value we find on the table we multiply it by two and if it's positive we say one minus the value we find on the table if it's negative the value we find on the table we multiply that by two and that's what we did on this and I guess we write on time every day we finish on time and that concludes our two hours of state any question comment next if there are no questions comments or anything then enjoy the rest of your weekend thanks Lizzie I will see you on Wednesday when we do assignment five which is based on study unit 10 and 11 which is chi-square and regression and I think because those ones we did them last it should still be in your mind as well and then we will discuss how we proceed in terms of the exam preparations because I know that now some people have already started writing the exams so we need to not overwhelm you with a whole lot of information maybe we can now decide to have only one session per week instead of two to accommodate those who are writing the exams who already started writing the exams otherwise enjoy the rest of your weekend don't be with me bye thank you bye bye