 in the next five minutes about, which is just to recap on certain things that you need to know or prepare yourselves. Other than that, then we will just address any comments, questions that you can ask anything relating to your assignment for, and then we can iron it out. Okay, so just to refresh your mind, before we start with our question and answer, remember your assignment four deals with two study units, study unit eight and study unit nine. Study unit eight is confidence interval, study unit nine is hypothesis testing. When it comes to confidence intervals, you just need to know and remember that sometimes in the assignment, they might just ask you questions relating to calculations. But when you go write the exams, they will, they can ask you questions relating to the content as well, like your concepts of critical, oh, sorry, not critical confidence intervals. You just need to make sure that you know both how to do the calculations and how and what are the basic concepts. So in terms of the confidence interval, the formula is always using the point estimate, which we use here, the sample statistics variables. So it's either going to be the x bar for the mean or for the proportion we use the p. Plus or minus, which gives you your lower limit, which is your minus will give you the lower limit, which is the something that you need to calculate first, and then the plus will give you the upper limit, which is something that you need to calculate after you've calculated the lower limit. And plus or minus your margin of error and your margin of error always remember that your margin of error is made up of your critical value times your standard error. And your critical value, depending on whether you're doing for the mean and the population standard deviation is known, that you need to know that you need to use the Z test. And if you doing where the population standard deviation is unknown, you need to know that you need to use the t test and then for the proportion is the Z as well times the standard error and your standard error also depending on the different section that we just spoke about. So these are the formulas that you always constantly need to always remind yourself that if your population standard deviation is known, we use your x bar plus or minus your critical value, which is the value you find on the table by dividing your confidence level. And remember your confidence level is one minus alpha by using your alpha value from your confidence level to find your critical value, which is the probability inside the table you go find your Z values. Know that your Z values are your critical being multiplying that with the standard error, which is your population standard deviation divided by the square root of N. And always remember that this is what we call the margin of error or the sampling error, which we can denote it by E as well. That is the margin of error. If they ask you to calculate the margin of error, it's taking the critical value times the standard error. If the population standard deviation is unknown, therefore they gave you the sample statistics. Then we use the mean, the sample mean plus or minus your critical value. Always remember this critical value, you're going to find it by using your alpha divided by two and your degrees of freedom, which is N minus one. So your critical value here, you will find it by going to the T table, critical values of T, using the probabilities at the top of the table. And remember it is those upper limit or upper tail probabilities that are closer to the table values and your degrees of freedom that runs down on your left. And for the proportion, oh, sorry, and multiply that with the standard error, which will be the sample standard deviation divided by the square root of N. With your proportion, then it will be your sample proportion plus or minus your critical value, which you will find on the Z table, times the standard error, which is your sample proportion times one minus the sample proportion. Divide by N. If they didn't give you the sample proportion, remember that you can always find the sample proportion by using the X observations divided by the sample size. And always remember this is your margin of error. This is how you will find the margin of error depending on whichever section it is. And that is in a nutshell, what you need to know about the confidence interval. Do not forget that the minus gives you the lower limit and the plus gives you the upper limit for all of them as well. And remember also, you can also use this table if you don't want to go to the table, the atrial table, Z table, especially for the Z. If you don't want to go to the Z table, you can use this table to find your critical values. When it comes to hypothesis testing, you need to always remember the six steps of hypothesis testing. Step number one, you need to be able to know how to state your null hypothesis and your alternative hypothesis. It is very important to know how to state your hypothesis statement, especially your alternative hypothesis. The sign on your null hypothesis, it's something that always contains an equal sign and most of the time it will just be an equal. So you will say the population proportion is equal or the mean is equal. But the sign that you put on your alternative hypothesis on your H1, on your alternative will tell you whether you're doing a one-tail test or you're doing a two-tail test. And that sign usually does not have an equality sign to it. So it's either you're going to have a less than or a greater than for a one-tail test or a for a two-tail test, you will have a not equal. And that is very important because it will help you when you make a decision or you find your critical values or you find your P value. Step number two, you need to be able to identify the facts given in your statement in terms of the level of significance, which is your alpha value, your sample size or your proportions or your standard deviations. Step number three, you need to be able to state whether you're doing a Z test or a T test. A Z test, either when you're doing for the proportion, you will state that you're doing a Z test. If you're doing for the mean and the population standard deviation is known, then you need to state that you're doing a Z test. If the population standard deviation is unknown, you're going to do a T test. Step number four, you need to find the critical value. Depending on the type of a test, depending on your one test or two-tail test, you need to be able to define how you will define your physical values. So let's assume that our critical value, the null hypothesis, they stated it is less than. Therefore, it means when you go find the critical value, your critical value will be on the lower side. And either for the T or the Z, you're just only going to use alpha value. And also when you go and find the hypothesis testing was for the greater than, your region of rejection will be on the greater than side. And your critical value, you're going to find it by either using your alpha field. If it was not equal, then you have two sides to make your decision. So your region of rejection will be created in two sides. So either it will be, oh, sorry, I forgot to put the minus on this one. It will be minus T alpha divided by two or it will be minus Z alpha divided by two. Or the side will just be T alpha divided by two and Z alpha divided by two, depending on which ever test you are doing. So always remember for a two-tail test, you will need, you will have two regions of rejection. And for a one-tail test, you only have one region of rejection. And this is very important as well, because when you make a decision, you need to be able to make the correct decision. Step number five, you need to calculate your test statistic, which is the same as your sampling distribution Z score three. Step number six, you make sure that you make a decision based on there are two ways you can make a decision. So now the first one, you can make a decision by using the critical value and your Z test statistics, which is what I have just done there, creating the critical values. You look at your test statistics and see where it falls and you make your decision. Another way of making a decision is by using the p-value. So now when you use the p-value, there are different things that you need to take into consideration. Now with the p-value, you don't have to worry about the critical value. You use your test statistics to find your value on the table. The first thing you need to be aware of is if it's a one-tail test, we're going to say if it's a, or let's start with a less than. If it's a less than, it's a one-tail test for the less than value. So you need to make sure that your Z, even if it's positive or it's negative, how you will find your p-value. So your p-value will be equals to the value you find on the table. That is the, for when your hypothesis-tasty or your alternative hypothesis-tasty states that the mean is less than or the population proportion is less than. If it is greater than and your Z is positive or your Z is negative, then your p-value will be equals to one minus the table value. So you just need to make sure that you understand the process of finding the p-value. The last one, it is when it is not equal. When we say in your hypothesis state that the mean is not equal or the population proportion is not equal. Then it is a two-tail test. To find the p-value, first we need to look at when the Z value is negative. When the Z value is negative, then your p-value will be equals to two times the table value, the value you find on the table. If your Z value is positive, then your p-value will be equals to two times one minus the table value. So that is very, very, very important to always remember that. Okay. These are the test statistics that we need to be able to calculate. As you can see that the formulas look almost exactly the same as what you did on the sampling distribution and you use your p-values. Only the p-value is only applicable when you're doing a Z score. So only for when the population standard deviation is known and also for the proportion, not for the T test. Also remember, sometimes they will give you words always in the back of your mind know how to translate the word into a mathematical symbol. Whether they're asking you for a greater than or exceed at most, less than, fewer than you always constantly need to remember all that. And that is it from my side. Are there any questions? Let's get to question and answer session. Is there something that you want to ask? Anything? I'm going to stop sharing. Are there questions? Remember today is question and answer. I have given you a whole lot. Is there something that you still unsure of that you want us to go through in detail today? Lizzie. Okay. For me, it's questions with regards to Apple. This is the study unit nine. Yo, I just can't get them right. Yeah, I was just asking if maybe we can do like two or three of that if you have on your notes because I didn't bring anything. Please. We can do that. We can find one or two questions. I just want to open. Just want to open one past. Tutorial letter or something. That we can use. Let's give you a second. Okay. I don't even know where to find a question paper. Oh, there we go. Found it. Let me share my entire screen. This is 2020 tutorial letter. I hope it has some questions. Yes, it does. You want questions relating to hypothesis testing? There we go. Confidence interval, confidence intervals. And here is one question. So let's see if we can answer any of these questions. Which one of the following steps is not a step in the hypothesis testing? Option one says formulate the null hypothesis and alternative. Step two, specify the level of significance. Step three, determine the test statistics, critical value and the value. Step four, determine the rejection rate region. Step and then option five. None of the above. Which one would you choose? Number five. That's correct. That will be number five. All of these steps are part of hypothesis testing. We just went through the steps now. You are able to determine the region of rejection by using your critical value. You are able to find the test statistics and so forth. And the level of... It's step number one where you can find your level of significance. Step number one, you need to always state your null hypothesis and alternative. Okay, so now let's go to question 10 of this. So question 10. What are the practices statistician is tasked with investigating whether the population mean of a random X is different from 100? She has the following additional information to help her with the investigation. Alpha is equals to 0.01. Sigma is equals to 3. X bar is equals to 99. And N is equals to 100. So they have given us the population standard deviation. Sigma is known. And from there, you can also even automatically speak to yourself and say because it's known we're going to use Z test. Okay, so they say consider the following statements or consider statement A to E below. So they have A, your null hypothesis states that the mean is equals to 100 against the mean of not equals to 100. B, the null hypothesis states that the mean is equals to 100 against the null hypothesis of the mean of less than 100. C, the level of significance is 0.01. D, A is a Z test statistic is applicable. So now let's go back and understand what they have given us in the question right there. But Charlie, a practicing statistician is tasked with investigating whether the population mean of a random variable is different from 100. What does that mean different from 100? What does that mean different from 100? It means equal or it's not equal because if it's either equal means equal means they are the same. Not equal will mean they are different. So that's what they are taking as they different will mean not equal. So they are different. She has the following additional information to help pair with the investigation, the level of significance, population standard deviation, the mean and the sample size. And they say consider all these statements below. What is the question asking us? We've went through the statement. I'm not going to touch on them again. Which statements are correct? So let's go back to each statement and check which one of this is correct. Based on the information, would you say A or B? Because A and B are stating the null hypothesis. Which statement from between A and B? Would you say it's correctly stated? Based on the information given is A, how you will state the null hypothesis and alternative statement for the statement. Or B, is it how you will state it? Which one? Isn't it A? I'm not sure. Based on the information, I just told you because I gave you all the answers. Based on the information between A and B, which one would be correct to represent the statement? A. A will be correct and B will not be correct. Because I already told you that it says your X variable, the population mean, they already told you that the population mean is different from 100. So different means not equal. If they said that the population mean is the same as 100, it would have been equals to 100. If they would have said it is less than 100, then we would have used the less than or if they would have said it's more than 100, we would have used the more than a greater than sign. So because they say it is different, we need to be able to read the questions and the statements to identify. So A is correct. So now moving to C, would you say C is correct? What is the level of significance? Pardon? C is correct. C is correct because your level of significance is 0,01. So C is correct. Are we using a Z test or a T test? I think we are using Z. We're doing a Z test because I already gave you the answer as well, because why are we using a Z test? Because the population standard deviation is given, right? So if the population standard deviation is known, we use a Z test. So G is correct, E is not correct. Now answer the question. Which statements are correct? A, C and D. It's only A, C and D which is option 3. You will choose option 3 in your exam or assignment, whichever one comes there. Based on the same information, because if I go back to the question, I'm going to assume that this information that is given in question 10, it relates to the next question. What will be the value of your test statistic? So now we know that we're using a Z, so therefore we need to calculate the test statistic, which is your sample mean minus the population mean divided by the population standard deviation divided by the square root of N. So let's go back. You can write the formula, so this will be Z state. You can write the formula down and then go and substitute the values. So we know what our sample mean is 99. Just substitute 99. Our population mean is 100 because it's what would have stated in the hypothesis. It's what is stated in the hypothesis, it's 100. And you just say minus 100 divided by the population standard deviation is 3 and the sample size is 100. So you just say 3 divided by the square root of 100. Then calculate 99 minus 100. Let's use the fraction. 99 minus 100 divided by 3 divided by the square root of 100. What is your answer before I press my one? Negative 3.33. Negative 3.33, which is option 1. The next question, find the critical value. What will be the critical value? So always remember, go back to the question, our alpha is 0,01 and we're doing a two tail test. So it means we're going to find our alpha divided by 2 by using Z. Or you can go directly to the table. Remember the table? The table we set because we're using the Z, you can use the table. So alpha of 0,01 divided by 2 and it will be Z of 0,005. Let's go to the table just for argument sake. Z of 0,01 it will be for a 99% equals 1 minus alpha is equals to 0,99. For alpha will be equals to 1 minus 0,99 which is alpha is 0,01 which is the same as the alpha that they gave us. So that is why hence I'm using a 99 confidence level and your Z value will be 2,58. Or you can come to this table, look for 0,05 inside the table 0,05. I will use that one and go out. We find 2,5 and then we go out at the top and the last digit is 8 which is 2,58 which is the same as this value that we have there. So you can decide whether you want to use the table or you want to use the extra normal distribution table. Which is option 2,13. It's asking you to find the P value. Always remember because we're using or we're having a two tail test, therefore our P value will be two times depending on the Z value. Now we need to go to our Z value. Our Z value is negative and since the Z value is negative we're just going to say times the table value. So now we know what our Z is. It's minus 3.33. Therefore it means we need to go to the Z table to find minus 3.33. So let's go to the Z table. We're looking for minus 3.33 remember minus 3.33. The first 3,3 we'll find on the left and the last digit 3 we will find at the top where they both meet. That is our P value which is 0,3043. Two times 0,00043. Which will be equals to, I hope I'm calculating it right. Let's see 0.0043 times 2. And if you have a calculator that does that just add one. And that will be 0,0086 which is our P value. So now we use the process of elimination. It's not that, it's not that. And it's not this, right? And it says the P value is less than 0,002. It's not that. None of the above is the answer for the question that we have. I don't know what they mean by either the two can be correct because it says the P value which we found it's less than 0,002 which is correct. Or it can either also be none of the above. I don't know what they refer to with that statement as well. It's one of those. Okay, so let's look at other more questions. Let's see are we still on? Yes, we're still on the same question. What is the decision with regards to the hypothesis and the conclusion about the population mean? We can use the critical value of this instance. Let's use the critical value. We found that the critical value is 2.58. We know what the region of the test statistics is. Let's make a decision. So let's draw our normal distribution and let's create our region of rejection. I put in our Z of minus 2.58 and on this side because we're doing a two-tail test. On this side we have the positive 2.58. So we know what our Z stat is equals to minus 3.33. So based on that information, are we rejecting the null hypothesis? If it falls here, we will reject the null hypothesis. If it falls on the red shaded area or are we not rejecting the null hypothesis? Where does your Z stat fall? Does it fall in the rejection area? Sorry, Rebecca was checking what the message was about. Where does minus 3.33 fall? It falls in the rejection area. It falls somewhere there because it's way past the minus 2.3. So therefore we reject the null hypothesis since the Z stat. It's less than the Z critical value and we know that our Z stat is minus 3.33. It's less than the critical value of minus 3.58. So now let's read the question. What is the decision? We reject the null hypothesis and conclude that the population mean is equal. So remember now our null hypothesis stated that the mean is equals to 100. The alternative hypothesis states that the mean is not equals to 100. So if we reject the null hypothesis and conclude that the population mean is equals to 100, is that true? That's not true because we are rejecting this. So therefore we say it's not equal, right? So we say we reject in the equal. That's what we said we are doing. So we reject the null hypothesis and conclude that the mean is not equals to. And we conclude by saying the mean is not equals to 100 because then the alternative is true. That would be the correct one. So we know that we are rejecting so the two statements are irrelevant. Okay, any questions? Then let's look for more questions. We never use words like accept, right? So when you see something like this, always know that it gives you a red flag. Okay, let's look at another question. Otale and Mabato are collaborating to test the hypothesis that the mean reading time for adults is less than. Therefore, yeah, we have less than less than 100. So this should be done less than 100. Consider the information from question five and assume at 5% level of significance. So it means our alpha here is zero comma zero five. Now, they say we must use the same information from question number five. So I need to go to question five to go find out what was that that was given. So write it out in question five. They told us the sample size is dating Mabato and Mabato the social scientists to get a random sample of 30 adults. So we know that our end is 30 write it out so that you can remind me when we get there with the sample mean and the sample standard deviation sample mean sample standard deviation. So it's your X bar and your S of 1890 and 1890, 18 and of date. Okay, so they also said question six, but we already know with 95% they gave us a level of significance of five. So we need to just remember 30, 90 and 18. So going to our question using the information from there. We know our mean is 90, our standard deviation is 18, our end is date and our alpha is that. So now the first thing that you need to do here is we before we can answer this question. Let's do the normal hypothesis testing statement, state number one statement number one. Let's see if we can state the null hypothesis and the alternative. The mean is equals 200. What will be the alternative hypothesis based on the information given the alternative will state that the mean is less than. 100 because that's what we will use whatever it's in the statement we will use answer this. And can you tell me are we doing a two tail test or a one tail test a one tail test or a two tail test. Anyone look at the sign. We are doing a one tail test a one tail test. A two tail test is when it is not equal is it's a two tail test a less than or a greater than is one tail test. It's very important to identify those because how you find your critical value and how you find your Z value. It's different. Okay. Also your P value would be different. Okay. Step number two. What is our alpha value? It's zero comma zero five. Our N is state. And we can also from here determine whether our population standard deviation is a node because they gave us the sample statistic. Step number three. Are we doing a T test or is a test? Is this going to be a T test? Or is a test. The population standard deviation is unknown. So when the population standard deviation is unknown we doing a T test a T test. Step number four. Let's find the critical value since we're doing a T test. We're going to find the critical value. Are we going to find it by using T alpha divided by two or just by using T alpha? Is T alpha because it's one tail test? Yes. We just going to do T alpha and your degrees of freedom which is equals to N minus one. So your T will be zero comma zero five and your N is state D minus one. So your T of zero comma zero five and the degrees of freedom of 29. So let's go find the critical value on the critical values of T table. Remember not to use all these values here at the top. We use the values right there. So we're looking for T of zero comma zero five and 29. So it means the degrees of freedom. Okay. Let's first find the alpha because when I scroll up your alpha value will be on this column. So we can go find our degrees of freedom of 29 and that is the value we're looking for. Write it out. It's one comma six nine nine one. So our critical value is one comma six nine nine one. That is our critical value. Step number five. We need to calculate the test statistic. T stat is equals to your sample mean minus the population mean divided by the standard deviation divided by the square root of N which is sample statistics. Your sample mean is nine minus your population mean always stated in the question. It's 100 over your standard deviation is 18 divided by the square root of N is state. Let's go and calculate. You have 90 minus 100 divided by 18 divided by the square root of state. What do you get before I give you the answer? No one wants to answer me. And the answer is minus. What did we get? Minus three comma zero four. Minus three. Yeah. So let's leave it to two decimal. It's minus three comma zero four. Zero four. Now we have our critical value and our T. Let's make a decision. Step number six. Let's make decisions. I don't know how to draw. I try my level best doesn't have to be 100%. That is very wrong what I'm drawing right now. We just need one side because we're doing a one tail test. So it will be only on one side because it says it's less than. So we have our critical value of minus and point six and one. That's our critical value for the less than and we can make our decision. So where does are we rejecting? So if it falls this side we reject the null hypothesis. So where does it fall? Does it fall on the rejection area? Yes. Therefore we are rejecting the null hypothesis at five percent level of significant. That's the other way you can do that. For a T test, it's very difficult to find the P, the P value. Unless if they give you the P value. So since we since we doing a T test, we're not going to worry too much about the P value. So let's see if we can answer some of these questions. They are asking us to find which one is incorrect. So let's start with number one based on the information that we just went through. Number one, is that correct or incorrect? Are we doing a one sided test? That is correct. Yeah. Number two, is the critical value 2.045. False. That is incorrect. The value of a test statistic. True. True. That's true. We reject the null hypothesis and conclude that the mean reading of adults is less than. 100. True. And if we were given the P value, we're also going to reject the null hypothesis because then if our P value is less than our level of significance, we would have rejected the null hypothesis. But also, yeah, they gave us also the wrong alpha value. So these are some of those things that I am not sure what your lecturer is trying to do with this kind of information as well. Because they gave us the level of significance and 0.05, not 0.005. Right. Can I quickly see question five please? The only information we need is this information here. It's the statement. Thank you so much. Okay. Cool. Let's see. Are there any other questions so that we don't really do the hypothesis test here unless someone also has another question. Okay, so let's look at another question on hypothesis testing. I think we're still on there, but this is of the proportion, never dealt with proportion. So we'll do the same on this question. We'll do all the six steps. I did once again. This time they want to determine whether a true proportion of ASD children in special needs school in the population is 75% or 0.75. Consider the information given in question seven and eight at 5% level of significance. So it means we need to go and find other information because we need to go to question seven and eight. At the moment we know what we're given here is the population proportion. That's the population proportion of 0.75 and our alpha value, which is the level of significance. So let's go to question seven and eight and see what's in store for us. Question seven. Mabato randomly select 100 children. So our N is 100 with ASD and found that only 70 of them are in special need. Seven, two of them X in special need. So even though they said we must look at question seven and eight where they just give us the confidence intervals in those questions. Everything we need is in the statement. So 100 and 70 is what we need. We go back to our statement and we can answer our question. So we know the other information they have given us is N of, I forgot now. N, 100 and 70, N of 100 and X of 70. So now it's, let's continue. Step number one, state denial hypothesis and the alternative. So based on this information, our population proportion reading this doesn't tell us anything about less than or greater than or at least or at most. So we can assume that this is a two tail test 0.75 and the population proportion is not equals to 0.75. So this is a two sided or two tail test, two sided test. Step number two, what is it that we given alpha of 0,05 our N of 100. Our P is X over N. So X is 70 divided by 100, which gives us 0,70. Step number three, we need to state whether what type of a test is this automatically for the proportions. We doing AZ test. Step number four, we finding the critical value. So our Z critical value. And based on the information since we're doing a two tail test, therefore our Z will be alpha divided by two. Z of 0,05 divided by two, which will be Z of 0,025, 0 and we know that this is the same as 1,96. If you don't believe me, let's go there on our Z value table on the negative side. We're looking for 0,025 inside the table. And there it is. You go out, corresponds to 1,9 and go out to the top, corresponds to 96, 1,96. Or you can use this table for a 95% alpha value is 0,05. Therefore, our Z critical value only for Z alpha divided by two. It's easy to find the critical values here. Alternatively, you can, for yourself here, your alpha values and that you can use to find the alpha divided by two. You know that it is 1,96. Now, step number five is to calculate the Z state. The Z state for this will be our sample proportion minus population proportion divided by the square root of our population proportion times one minus population proportion divided by n. Substituting into the formula. Our P, we calculated it, it's 0,70 minus our population proportion is something that is stated in the hypothesis. She's 0,75 divided by the square root of 0,75 times one minus 0,75 divided by n. And it's 100. Okay. Do the calculation. Put them on my side, 0,75 divided by the square root of 0,75 times 0,75. And that is equals to, do you also get the same? Minus 1,15. Let's make a decision. Do we have a P value? Do you have a P value? Let's make a decision based on the P value. Let's not use the critical value in this instance because we did use the critical value. So our P value, we look at our test, it's negative. We know that we're doing a two sided test. So it will be two times the table value. Therefore it's two times and then we go to the table value. On the z table, we're going to look for minus 1,15. Let's double check that I have the right numbers, minus 1,15. So we go to the z table, we're looking for minus 1,15. Let's remove the last ink. 1,1. We need 1,1 on the left and we need 5 at the top. That's the last digit. This is the column we're looking for. So where they both meet, it's 0,12,51. 0,12,51. And 3 times 0,12,51. Double check, 1,151, that's correct. And equal 0,2502. I'd say yeah, it's 0,2502. What else do we know? So we have our p-value, however the decision, let's start with the rule. The rule says if the p-value is less than alpha, we reject the null hypothesis. That is the rule. Our p-value is 0,25,02. Our alpha value is 0,05. Therefore our p-value is greater than and therefore we do not reject the null hypothesis. So based on all the statements, now let's go into our answer. We're looking for the incorrect statement. Statement number one. Is it correct? Yeah or no? It's correct. Statement number one is correct. Statement number two is our critical value, 1,96. Yes, correct. That is number four, step four. And question number three, the value of our test statistic is minus 1,09. Incorrect. That is incorrect. Our p-value is 0,2502. Correct. And the last statement, are we rejecting or not rejecting the null hypothesis? We do not reject. So we do not reject which is correct. So as you can see that in all the steps of hypothesis testing, you need to know how to do them in order for you to answer any of these questions. Whether it's in the assignment or in the exam, you might find almost similar type of questioning that you need to know. So you need to know your six steps of hypothesis. You need to know how to state your null hypothesis correctly based on the information given. And the only way to know how to do this is by practicing. You cannot just come through the class and listen to me or watch the recordings and not practice and find questions and see if you are able to look at the question and identify certain information from that question. So you need to practice and practice and practice. So other than that, this is not part of today's session. It's next month's session. Are there any questions we left with? It's almost like 17 minutes. Any questions? No more questions? You can look for another question. I've got plenty. Because in the previous years, this used to be a semester module. So you will get first semester questions and second semester questions. Unless in this question paper there are the same. And then the same. I'm not sure if we did questions like this previously. I think we did. We did. So let's look at something different, something totally different from the hypothesis. Suppose we take a sample of 200 Facebook profiles and found that the four of them are ghost profiles. And this is confidence interval question. Because I can see from the answers that we have. So what have they given you? They've given you the sample size of N of 300 and your X 34. What is a 95% confidence interval for the proportion of Facebook ghost profile? So we also need to go find the critical value by using our level of confidence level of 90. The confidence level of 95, which means our alpha is 0 comma 0 5. So based on this, and since we're doing proportions, then I know that the formula will be plus or minus. So that critical value of alpha divided by two times your standard error, which is your P times one minus P divided by N. So we need to go and find P. P is X over N. What is my F? It's 34. What is my N? N is 200. Calculate. Give me the answer. 0.17 is 0.17. So let's go find the critical value that alpha divided by two. By now you should know what that is of 0 comma 2 5 0 0 comma 0 2 5. What is the critical value? I think it's 1.96. Some of this you should know them by heart instead of going there and there, especially for a 95% confidence interval. Or when you're doing hypothesis testing, as long as you do alpha divided by two, you always for 95%. Always remember that it's 1 comma 96. They like 95%. Since we have our critical value in our P, let's go and calculate or substitute our P of 0 comma 1 7 plus or minus our critical value of 1 comma 96 times the standard error square root of 0 comma 1 7 times 1 minus 0 comma 1 7 divided by 200. I can ask you to calculate the margin of error or the standard error. The sampling distribution, which is the square root of action 0.17 times 1 minus 0.17 plus bracket divided by 200. That is the standard error. Multiply my standard error with my critical value of 1 comma 96 and I get the whole number. I can write the whole of it 0 comma 0 5 2 0 comma 1 7 plus or minus 0 comma 0 5 2. While I'm still in the problem mode, I will need to write all of it 0 comma 6 0 6 0 0 4 9 9 4 4 9 9 4 and then you will need to do the upper limit and the lower limit. So first start with the lower limit and then go to the upper limit with this nicely and then go find the lower limit. So do the minus first and then the plus first second 0 comma 11 8 0 comma 11 8. There are four decimals. Okay, yeah it's 0 comma 11 7 9. 0 comma 11 7 9 and on the other side. 0 comma triple 2. 0 comma. 1. Right. Triple 2. 1. 2 2 2. Triple 2 1 yes. Probably the answer would have been option 2 so let me just double check here and less than the rounded of quickly 1 7 minus 0.0 5 2 1 equals. If I use the whole number 0.17 minus 0.0 5 2 0 6 0 0 4 9 9 4 9 6 or something. Okay, I still get the same. If I use the calculator to do the entire 1 0.7 minus 1 comma 9 6 times the square root 0.17 times 1 minus 0 comma 1 7. 200. Still get the same. Okay. And if we change the minus to a plus and we get 222. I don't know why they have those numbers anyway. So every time you see on my slides there are wrong values. This is where I get the wrong values as well. Just as a disclaimer. So really see on number 5 it says none of their path. I don't know maybe if the answer was that one of none of their path. Not really this last. This last option. Usually it is there to confuse you. Unless in this assignment they also did have an error to correct it. But the option that was submitted was option two for the backing of your scripts or we ever wrote this assignment. We are left with seven minutes. Let's see. This one. They are asking you to calculate. Only the upper limit but using a 99% confidence interval. So let's look at this. We were using a 96. So now we're going to use a 90. A 99 because I'm lazy to go back and calculate 499. The critical value is 2.58. So you do the same P plus or minus. Z of alpha divided by 2. That means the standard error of P times 1 minus. P divided by 200 divided by n. 0 comma 1. 7. Actually you don't need the minus in this one. You just need one thing. Plus 2.58 times the square root of 0 comma 1 7 times 1 minus 0 comma 1 7 divided by 200. And that should give you your upper limit. See on this one if we get the right answer. 0 comma 1 7 plus 2.58 times the square root. I got 0 comma 1 7 5 4. 0 comma 1 5 4. I mean 0 comma 1 7 5 4. Impossible. What did I do? Just double check. No, no, no. I was wrong. I was wrong. I left something. I left something. I see. So the answer is option number 4. Yes. The answer will be option number 4. So other questions relating to hypothesis testing. This one. They say question 9 to 11. I know that we left with four minutes, but let's see if we can answer this without doing all the six steps. We're going to answer it based on the information that they want us. So on this question they've given you the practicing statistician wants to test the hypothesis at alpha of 0 comma 0 5 or 5 percent level of significance. The null hypothesis states that the population proportion is equals to 0 comma 2 5. The alternative is not equals to 0 comma 2 5. Therefore, yeah, we're talking about a two sided test. From a sample of 100, our P, which is our population proportion is 0 comma 2 2. The question here he says calculate the test statistic. So the test statistic is that of your P minus your population proportion divided by the standard error, which is your population proportion 1 minus the population proportion divided by N. So we just substitute P they give you 0.22 minus your population proportion 0.25 because it's always stated in your hypothesis divide by a square root of 0.25 times 1 minus 0.25 divide by N. And it is 100. Calculate and give me your answer and I will give you my answer. You also get minus 0 comma 6 9 is minus 0 comma 6 9. And then question 10 says we need to go find the P value. Easy when the Z value is negative, right? Because then our P value for a two tail test. It's two times because the Z value is positive. It will be two times the table value. So we need to go to the Z table and look for 0 comma 6 9. Minus 0 comma 6. This minus 0 comma 6 9 is the last column. So it's easy. The answer is 0 comma 2 4 5 1 2 times 0 comma 2 4 5 1. Let's double check 2 4 5 1. That's the answer. And you just take your calculator and calculate that would be 2 0 9 4. Let's double check. I was counting in my head 0.24 5 4 times 2 0.4902. It's none of them. Blah blah blah blah blah. So the answer is option 1. It's typing error probably. I don't know. I can't say 0 5 1. And the last question following the P value would have been probably make a decision as much as I thought. Consider the decision and reasons with regards to practices that is decision hypothesis testing. So now they've given you 1 2 3 4 statements. So we can use one of them. Let's use the P value. So let's use the P value and your alpha value. So your P value alpha is that. So the decision. I'm going to write this here. So we know what the decision is. The decision rule says if the P value is less than the alpha, we reject the null hypothesis. So now our P value is 0 comma 4 9. That's what we have 4 9 0 2. And our alpha value was 0 comma 0 5. Right. So we know that our P value is greater than. Therefore we do not reject the null hypothesis. That is for the PV. Our critical value. So for the critical value. We know that it is a two tail test. So we'll have two regions of rejection. And our alpha value since it's 0 comma 0 5. It will be 1 comma 9 6 on the side and 1 comma. 9 6 on the other side. But we have our Z score of 0 comma 6 9. So where will it fall? It will fall some way inside. There and hence we do not reject the null hypothesis. So consider the following decisions and reasons with regards to practicing the hypothesis testing. And remember that question 819 11 uses the same information so we can use that. So both questions we do not reject the null hypothesis. So yeah, it says to reject. Yeah, it says to reject. We should not be looking at that. But let's just for argument sake read the questions and answers and see. We reject the null hypothesis because the P value is less than alpha. We not, right? That's not correct because we do not reject. We reject the null hypothesis because your Z value is greater than your absolute. Those are absolute. It means they are positive positive critical value. So your test statistic, your positive absolute critical value is greater than your positive critical value. We know that it is not greater than it's less. So we not rejecting the null hypothesis. Then we are left with these two statements. So both of them are correct. This statement, the first part of it, correct? Because it reflects what we just said. We do not, we do not reject. So let's read the whole sentence. We do not reject the null hypothesis because the P value is greater than alpha. That's true, right? P value is greater than alpha. That's correct. Number D. We do not reject the null hypothesis because the test statistic critical value, absolute value, it's less than the absolute value of that. That is correct as well, because if this is positive, that is positive, the positive will also, we will still be the absolute value. We will also be not rejecting the, we will not be rejecting the null hypothesis for the positive side, right? Because the answer was 0,69. So the positive test statistic will still be on, they do not reject the null hypothesis. So which one of the following statements are correct? We have C and D, which are correct. So only C and D are correct. Anyway, that ends today's session. Enjoy the rest of the weekend. If there are more questions, let's chat on WhatsApp. Are there any questions or comments? If none, then have a lovely Sunday and good luck with your assignment four. But remember, we are always a little on WhatsApp to assist if you have any question or you still unsure of certain things. Thank you very much. Bye. Thank you. Thank you. Bye.