 And welcome to another session of your basic statistics literacies. Please complete the register. And remember if you have any technical queries, send an email to CTNTAC, any content related issues you have with your module, you can send the email to eboy at unisa.ac.za and copy CTNTAC from that email. So today's session, we're going to continue from where we left off the last time we did sampling distribution that was in June. So this week we're going to cover two sessions because last week we didn't have a session. So I feel we need to cover what we were supposed to do last week and what we're supposed to do this week. And this will enable you as well to help you with doing your assignments. Okay, so in the first hour we're going to look at confidence intervals. And then in the last hour of the session, we're going to look at the hypothesis testing. So we will do this, do a lot of activities and I will also give you a chance to do some activities as well. And then in September, those will be the topics that we are doing. Those who are doing 1501, I don't think you do chi-square test. But if you do, then we do have a session next week suddenly. Otherwise, if you do not do chi-square test, then I will see you when we do regression line. Which will be on the 10th of September. Then after that, then we can start doing exam preparations after those two sessions. Good morning. We almost done. Good morning. How are you doing, Mrs. Lee? I'm good. Hi, Justice. How are you? Yes, actually we do chi-square test as well mostly. Oh, you do? Yes. I think there is one of these that you don't do is the linear regression. We do that as well. Okay. Then I will see you on the third and the tenth. Okay. Thank you. Yeah. Then we will be done with the content and then we will start doing exam preparations. And depending on when your exams are, then we will get that as well. And how we can set the sessions to be very fruitful for you guys. So, when it comes to confidence intervals, by the end of the session, you should learn the basic concept at the high level. I'm not going to touch on the nitty-gritty in terms of a lot of things when it comes to confidence intervals. So, I'm just going to do the skills and type of questions that we need to focus on how you answer the questions relating to confidence interval. But since we're building or when you answer questions on confidence interval, you construct confidence interval and it can be for different things. There will be three ways that we talk about confidence interval when the population mean and when the population mean is known and when the population mean is unknown and also for the population proportion. So, we're going to look at those ones. Like I said, for confidence interval, we're going to look at three sections of it. And when we do confidence interval, we're trying to see whether your point estimate, especially your parameter estimate, does it fall within the confidence intervals or limits. And you will have your lower limit and your upper limit of that. And your point estimate will just be that one single value that we want to check. And usually for the calculations, we're going to be using the sample point estimate because we don't always know what the population parameter point estimate is. Hey, so, in general, the formula will always look like this. This is the formula for confidence interval. You will have your point estimate plus or minus, which tells you your lower limit and your upper limit, the minus represent the lower limit and the plus represents your upper limit. Plus or minus the critical value, I will tell you how to find the critical value because we're going to use two tables, the z table and the t table. So finding the critical value for those three tables, you need to know because the method is different. And you need to multiply that with your standard error and the standard error we've learned about the standard error in sampling distribution as well. So you go into multiply your critical value with your standard error. And your critical value and the standard error when the product of these two creates what is called a margin or a sampling error. It's called a sampling error or it's called a margin of error. So margin of error is your critical value times your standard, your standard, that is what a margin of error is. Okay, so let's look at the different formulas that you need to know in order for you to answer any confidence interval question. So if we do confidence interval for the population mean and the population standard deviation is known, we're going to use our point estimate will be our sample statistics mean. Plus or minus the critical value where we will find it on the z table and your level of significance will be given to you. So we will use our alpha because the level of significance is from your confidence level and your confidence level is one minus. So a confidence level a confidence level is one minus alpha and this alpha value that we find from the confidence level we use it to go and find the critical value and this alpha value is also what we call the probability value. So therefore it means when we do alpha divided by two, we're going to use this value here as our probability on the table from a. Elite on the table and go find the Z and that will be your Z values on the table and you multiply by the standard error and we know that the standard error when the population standard deviation is known, it will be your population standard deviation divided by the square root of N. And for the population mean when the population standard deviation is unknown, then it means they would have given you the sample standard deviation and then the formula will be the point estimate plus or minus the critical value. Now the critical value for the T table. It's different for that it's alpha divided by two and go to the standardized normal distribution table and find your Z value. So you will look at the probability inside the table and go find the Z value outside with the T table. The critical value we find it by using our T alpha divided by two, which is your probability value, but also we need to use the degrees of freedom, which is N minus one and our N is our number of sample size minus one. So on the table, then it means we're going to have your degrees of freedom and the probability values in order for us to go find the critical. And we will look at that table just now. And you multiply that with the standard error, which is your sample standard deviation divided by the square root of N. For the population proportion, our point estimate is your sample proportion plus or minus the critical value, the same critical value that we use on the population mean for the Z test. You use it for the proportion, so proportion and when the population standard deviation are known, they use the same Z value. So it's your Z value times the standard error. Now, because we're not given the population parameters here, so our standard error, we're going to use the sample parameters or sample statistic, which is your sample proportion times one minus the sample proportion divided by N. And we know if they haven't given you the sample proportion, they would have given you the observation satisfying that in order for you to calculate the proportion. And that those are the formulas that you need to know in order for you to answer the question. And therefore it means you need to be able to identify the facts given in the question so that you are able to know which formula it's applicable for you to answer that. But that does not, and then let's look at how we find the critical value. So now here is a short table. You can take this and put it somewhere where you can use it for reference, especially now since you're writing online, you can remember and memorize some of these values as well, which makes it easy. So when you practice, the most commonly used critical values are 1995 and 99 confidence levels. So how do we find the Z critical value? So in the question, they will ask you that for 95% confidence interval, construct this confidence interval, right? So let's start with 95%, which is the most common one equation. So when they talk about 95%, remember I told you that the confidence level, I said the confidence level is one minus alpha. Therefore they are saying in this 0,95 because we're talking about 95% confidence level is equals to one minus alpha. And that we can solve for alpha and bringing alpha onto the other side, it becomes positive and bringing 0,95 to the other side. It becomes negative 0,95. Therefore our alpha value will be equals to 0,05. And that is our alpha. And in order for us to go find the critical value, we said we use Z alpha divided by two and our Z alpha is 0,05 divided by two. Therefore it is Z of 0,025. Then I said this is the probability value. So if you go to the Z table, if we go to the Z table and we look inside the Z table and it will always be on the negative side of the Z table because it contains the probability of small values. So we're going to go inside this table and look for 0,025. Because our table is four decimal, I can just add another zero at the end. So 0,2282202212, 0,0207202. Maybe I need to go down. 0,0256, 0,0250. Then I have found it. I need to go inside the table. I found the probability. Then I must go outside to go find my Z value. So you always first start with the left hand side. And you will find that it corresponds to minus 1.9 and go up, up, up, and it corresponds to six. Therefore, our critical value will be, because our critical value is our Z value. Our Z value is 1,96. We can just ignore the negative in front or we can add the plus because it's a plus or minus on the formula. So all we can just ignore. When we read the Z value on this table, we just don't read the negative side. It will not make any difference when we go to the formula. So we don't read the sign. And then we go to our critical value. It's 1,96. As you can see there, it tells you that for a confidence level of 95, our confidence level, because we know that it is 1 minus alpha of 0,95. It corresponds with a critical value of 1,96. So you can do the same with other values as well, or 99. And these are the values that I gave you here. And the only exception, it is with 90%. With 90%, the value that you will find, so our alpha value for 90% will be 0,10. So therefore, alpha divided by 2 will be 0,10 divided by 2, which is equal to 0,05. If you go to the table and go look for a value that corresponds to 0,05, you will notice that there are two values that are located between the two values. So 1, 0,05 and 0,50. The difference is 5,5. So with that exception for this one, we're going to take 4,5 as a, we're going to read both of the values, 4,5 as well. Only for 90%, it has 3 decimal. The rest of the other values only have 2 decimal. So that is, hence I say, you need to remember all this or memorize them or keep them safe somewhere where you can always refer back to them. Otherwise, you need to use your Z formula or your Z table to go find the critical value. Okay, let's look at an example on how we answer questions relating to confidence interval. Africa check is interested in activity of fake news tweets. From a sample of 50 tweets, there are 100 impressions on average. Assume the fake news tweet activity is normally distributed with the population standard deviation of 25 impressions. What is a 90% confidence interval for the population me? So now going back reading what the question or making sure that I understand what the question is asking me. The question is asking me to construct a confidence estimate of 90% for the true population of mean. And what have they given me? Identifying my facts, they have given me my sample size of 50. They also say on average, which means this is my mean X bar. My mean is 100. Assume that the fake news tweet is normally distributed with the population standard deviation. So they gave me my Sigma. So it means Sigma is known. And when Sigma is known, I use the Z. And that helps because now I can identify immediately from here the formula that I need to use Z alpha divided by two and times Sigma divided by the square root of N. So because they gave me all this information, I am able to do that. The first other thing, 90% confidence level, which is 0,90 equals one minus alpha. Therefore, alpha is one minus 0,90 and alpha is equals to 0,10 and alpha divided by two would be equals to 0,10 divided by two, which is equals to 0,05 and I can take my Z alpha divided by two, which is Z of 0,05. Go to the table and go find my critical value. So I can go there to the table. If I want to take the long route, come to the table, come find my 0,05 and I know that it corresponds to 1,645 because it's one of those exceptions and then I'll use that. So I know that it corresponds to 1,645 because I have saved this table somewhere and I can come back and refer to it. I can just come here and get the Z value and I'm going to substitute that into my formula. So you can do it either way. So our mean, we said it is 100 plus or minus our critical value, we did find that it was 1,645 multiplied by our standard deviation. It's 25 divided by the square root of n, we found that n is 50 and we can solve this. So this I can do this manually plus or minus and first start on the left hand side to calculate my margin of error. So let's calculate the margin of error. So we start with what is inside the bracket, which is 25 divided by the square root of 50 equals and I get the answer as 3.53 some number. Multiply the answer that I get with 1.645 and equal and the answer I get is 5,8159. I must write the whole number to remember what I said. 5,8159, it doesn't end there because I'm still in the problem mode. I must just continue to write all the values. 5, 9, 5, 3, 2, 7, 5, 5, 3, 2, 7, 5, 5, 3, 2, 7, 5 and that is my first definite and I can say this is 100 minus 5,8159, 5, 3, 2, 7, 5. That is my first side, my lower limit, my upper limit will be 100 plus 5,8159, 5, 3, 2, 7, 7, 5. So I have my lower limit and my upper limit. I need my calculator to calculate the whole equation. So 100, you always start with the lower limit, 100 minus 5.8159, 5, 3, 2, 7, 5. Equal and the answer is 94.184, 94.184. I'm going to leave it at 4 decimal because my answers here are 4 decimals. I'm going to do the last upper limit. My upper limit, I just need to go and change the sign from minus to positive, which is plus and the answer is 105.8160 because I need to round it off 105.81595, 595, if I round it off, the number to the right is greater than or equals to 5. So I must add 1 to the number to the left. And if I add 1 to 9, it becomes 10. I carry 1 and therefore it means it's 6. And the answer will be option 1. Easy, right? Yeah, I know Cecily. Yes. Yeah, on this mean plus minus. This is that table multiplied by alpha over 10. No, this, are you referring to this part? Yeah. That is your z over 2. This is the z over 2, which we use. It's how you're going to find your critical field. We're looking for the z value and your alpha over 2 is the probability on the table that will tell you where you need to locate the z value. It's one number that we need, which is the 16, 1,645 that we found. So we don't multiply z by alpha over 2. It's z alpha over 2 is one number. It's just a formula. Okay. We're just going to find one number with that. Okay, so let's look at another example. When we are not given the population standard deviation. Mabato, the social scientists took a random sample of 30 adults with autism spectrum disorder and found that they are really in time to be normally distributed with a sample mean and a sample standard deviation of 90 weight per minute and 18 weight per minute respectively. So when they say it respectively, then it means the first value that they mentioned corresponds to the first thing that they mentioned. So the sample mean will be 90% so that it means they gave us our x bar and the standard deviation, which means this will be our s because our statistic for the sample standard deviation is represented by an s. Remember that for the population parameters, we use the group letters for the sample. We always use the romance letters, right? So the question also, what else is given? They have given us our n, which is 30. They are also asking us to find a confidence interval at 99% confidence level and you can go into the same 0,99 is the same as 1 minus alpha. Therefore alpha will be equals to 0,01 and alpha over 2 will be 0,01 divided by 2, which will be 0,005, right? So therefore it means our set of 0,005, we can go and find it on the table. I'm going to use the table for now and not refer to the confidence intervals that I gave you. So we're looking for 0,005, so 0,00. This one is 51, it's way past them, this one is 59, so I can use this number. I'm going to use this number because I know already what the critical value is. It's 2,5 and when you go to the top, response to 8. So my critical value will be 2,58. That is the critical value and that is the same as the critical value that we got from here. 499% is 2,58. So I know that this one will be 2,58, that is our z, our z alpha divided by 2. So the formula that, ah, who can tell me what is wrong with what I just did? Because we are given the standard deviation, we cannot use the z, that is where I am wrong. We're not using z to find the critical value. We need to use t because why I'm making a mistake is because I didn't even go and look at the formula. So you need to first identify the formula, the formula is the mean plus or minus your t alpha divided by 2 and the degrees of freedom as divided by n. And if I did this first, I wouldn't have gone to the z table and go find the wrong information. I would have noticed that I need to use the t critical value. So the t critical value, we find it by using t alpha over 2 and the degrees of freedom and we know that our degrees of freedom is n minus 1. So already I have calculated what my n, my alpha over 2 is n minus 1, our n is 30. So n minus 1 will be 30 minus 1. So our t of 0,005 and the degrees of freedom of 20, 29. But let's go and find that critical value. So you need to pay attention to those small details. They can give you a wrong information if you use the wrong thing. Because here we would have used 2,58 and we would have gotten the answer wrong. So we need to go to critical values of t. There is a table that reads critical values of t and you don't have to worry about the cumulative probabilities. We only looking for the values closer to the table under the upper tail area. And here is your degrees of freedom on your left and at the top you will find your alpha divided by 2 which is your probabilities. So we looking for 0,05 which is the last column and we need to find the degrees of freedom here of 29. Remember we looking for t of 0,005 and 29 degrees of freedom, so 29. And we know that the last column is our 0,05 where they both meet that is the freedom that we looking for. 2,7564 Our critical value here is 2,7564. So we going to substitute into our formula. The mean is 90 plus or minus 2,7564 times our standard deviation is 18 divided by the square root of n. And that will calculate rest of the margin of error. The margin of error will be 18. I used my calculator to its capabilities. 2,7564 times 18 divided by the square root of n. Close bracket and equal 9,0584 90 minus 9,05 9,058457 Not nice to toggle between. 45 And that is our lower limit, our upper limit 90 plus. I'm not going to write all of it, I'm just going to dot dot after that. 84 dot dot dot dot dot it represents the rest of the other values. And because I have lying on my calculator, which has all the capabilities, I can just use 90 minus 2 comma. That gives me my lower limit as we can just look for the answer. 8,09415 8,094 I guess my answers here don't correspond with this but this might be the right one. And I'm going to assume that they used 258 to answer this one. Which is why the answer there is not correct, but number two would have been the closest. I'm just going to double check. I'm just going to double check 258. No, they used, you can see that they used two of them. The wrong one, because the answer they would have assumed that there is the answer. But yeah, that is not the right answer. The right answer should be, we should choose number four. 27564 Okay, so on this question also they had a mistake there because this is not 29. We will fix the slides. Okay, and then let's look at the upper one. Just change the minus to a plus and 0 comma 99 comma 05 84, I will choose option two for this one as well. So we can always change the slides numbers. So that would have been their answer. So that is how you will answer when you are given the sample standard deviation. Okay, I'm just not going to do the other one. I just want you to do this exercise. I'm going to give you time to do this exercise. And yeah, you are given the population standard deviation. So it means you're going to use the formula. X bar plus or minus the alpha divided by two times the population standard deviation divided by the square root of N. And you are given 95% confidence level. So therefore your alpha is 0 comma 05. And your alpha divided by two is 0 comma 025. I'm just going to give you your critical value, which is alpha divided by two. It's one comma nine six based on what we know. So just answer that question. You're giving us alpha divided by two is one comma nine six and the other one. No, just the only thing that I'm giving you otherwise then try and answer this. Find this confidence in cover. Okay. Please also remember to complete the register. Are we winning? Calculating. Let me know when you're done. I'm done on my side. Others? I only managed to punch the mean. And the data is that. And the. And the number of what you call. That's those numbers. Okay. Some of samples. All right. That one which is alpha over this and is giving me a problem. I've given you this. You just need to substitute that value. That is your Z alpha divided by two. It's one comma nine six. This whole thing. It's one comma nine six. I have done this. The alpha there on top of this. Oh, you mean the sigma? You mean this? Yeah. Sigma. Okay. All right. So let's look at this. What are you giving? According to me. I think the mean is hundred and twenty. The mean is hundred and twenty. Yes. Right. Your N. My N is eight. Your N is eighty. Your standard deviation. That is the sigma that they gave you there. Right. And 90. Yes. That represents standard deviation. Yes. So at the 95% confidence interval, we know that it is one comma nine six. So just substitute the values that you have onto the formula. You said your mean is hundred and twenty. It's my pen writing like this now. It's boycotting. Okay. So your mean is hundred and twenty plus or minus. Or we can also even just split it. Hundred and twenty minus one comma nine six times your standard deviation of twenty divided by the square root. So you can write it like this. One comma nine six times your standard deviation of twenty divided by the square root of three. That is your lower limit. Your upper limit. One twenty plus one comma nine six times twenty divided by the square root of eight. What was the answer that you got? Your mean is one one twelve comma eight four three. Okay. So that will be one of the answer and the upper limit. Anyone? Upper limit? One two seven point one five six nine. One two one two seven point one one five six nine. Which is not that anyway. I don't know. This comes from your past your tutorial letters and exam papers and all that. I can just double check as well from my side to this way. Let's do that. I can use this calculator. So we have one twenty. I think it is number one. Is it is it five? No, it's number two. Okay. Let's double check that I will do it from here. One comma nine six because then if we have multiple answers I will have to do it from my side just to double check. We have twenty divided by the square root of thirty. It is number two. He is correct. Yeah. And equals one one two comma eight four three also number. The only one with eight four is number two. And then we're going to change minus two eight plus twenty. Did I delete everything? I'm twenty plus one comma nine six times twenty divided by the square root of eighty because so also we have an issue with the CMATs. So it means some some way somehow they grounded off quicker on on one of the questions when they were answering this when they were doing practice exercises. The answer will be option two. So let's look at the last one. Which is the proportion. So remember for the proportion. If they didn't give you your sample proportion you can calculate it because they would have given you your observation that satisfies that. So let's read this. Suppose we take a sample of two hundred Facebook profiles and found only 34 to be ghost profile. What is a ninety percent confidence interval for the proportion of Facebook profile? So here we can go back. We also still need to calculate a confidence interval at ninety five percent confidence interval and by looking at this since they haven't given me the mean and the standard deviation I can assume that I'm doing the proportion. So P plus or minus Z of alpha divided by two times the square root of P one minus P divided by N. So I need to go and find out what I'm giving. I'm giving N which is two hundred. I'm giving X not X bar X. I was highlighting the number. So this is just an X. So since they didn't give me P I can calculate my P by using X over N that is 34 over two hundred. What is 34 over two hundred? 34 divided by two hundred is zero comma one seven. Zero comma one seven that is my P. Then I can just substitute into this. I know that my Z alpha over two for Z of zero comma zero five over two which will be one comma nine six. It's the same for a ninety five percent confidence interval. The Z value is one nine six. We know that because we can use this table. Remember that table. Ninety five. This is only for the Z values. Ninety five is always one comma nine six. So just substituting the values zero comma one seven minus one comma nine six. Because I'm going to do lower limit and upper limit times. I'm just going to put it in the bracket times zero comma one seven times one minus zero comma one seven. This bracket divide by N is two hundred. Close bracket zero comma one seven plus one comma nine six times the square root of zero comma one seven times one minus zero comma one seven divide by two hundred. Close my limits. Okay. So let's go find the answer. Zero comma one seven minus one comma nine six times the square root of the fraction. Zero comma one seven times one minus zero comma one seven close bracket divide by two hundred. Then go out out again and close bracket and equal my lower limit is zero comma one one seven nine. Which is this one. I can go into my limit. Since I have my lower limit. Upper limit class equals zero comma two two two zero six the rounded off two four decimal is two one. The answer will be four. Happiness. Are we happy? Are we good? Are we good? Still good. Okay. I have some exercises as well. Exercise two. With exercise two, you can do it on your own so that we can then move to the hypothesis testing. So let's look at what you are given so that you are able to answer this question. Exercise two says the following results were calculated. You can take a screenshot while I am explaining the following results were calculated from the data with the mean of 10 and the variance of nine and the sample size 16. Now reading this whole sentence from the following results were calculated. You should already realize from here they're talking about this was calculated from the data with your X bar already when they give you your X bar which is the mean. It tells you this is the sample right the sample mean and the variance. So you need to also say because the X bar and the variance comes from this same data already make that assumption that this is your sample variance. And now that's the other thing that you need to take into consideration. This is a variance. So it means it's a split. Right. So it means in order for you to substitute into the formula, you will have to do certain things. You have to find your standard deviation. This is the variance sample variance and your N is 16. So they just wanted to confuse you right there. But all what you need to do is know that you're using your confidence level of 99 and know that your alpha will be 0,01. The formula you will need to use plus or minus T alpha divided by 2 and the degrees of freedom which is N minus 1 time your standard error. So you need to find the S which it means S will be the square root of S squared. You will need to go find the critical value on the T table using your alpha of 0,01 and substitute and find this. That is one that you can do on your own. If you are struggling, we can discuss on WhatsApp. The last one is the proportion. So they also give you in the city with 25,000 people, a random sample of 600 revealed that 120 opposed the reelection of their mayor. Now, one thing that they want also to confuse you is give you so many numbers of which the first part is irrelevant to how you answer this question. There's a lot of information. The only thing that is important is your N and your X. And remember you can find your X over N and you need to use your P plus or minus Z alpha divided by 2 because it's alpha over 2. Remember the special exception Z over 2 for a 90% it's 1,645. You need to remember always remember that Z over 2 times the square root of your P times 1 minus P over N. You can do this as well on your own. And if you are struggling, you can have a discussion with me on WhatsApp. Okay. And that concludes Confidence Interval. Now let's move on to hypothesis testing unless if you have any question. And always remember, right, because I'm not touching on lots of other things. Remember that any value after the plus or minus all this creates your margin of error, right? Margin of error or sampling error. So it's your critical value times your standard error that gives you the margin of error. Whether you're doing it for Z or for this all that is your margin of error. Critical value times the standard error gives you the margin of error. And always remember that the last part is your standard error. So this is your standard error. This is your standard error. If they ask you to calculate those things, you just need to always bear in mind. And always remember the plus is your upper limit. If they ask you to calculate the upper limit only, always remember to only do the plus side only. If they ask you about the lower limit, you know that you only need to calculate the minus side. Other than that, there's not much I can tell you about Confidence Interval. Sorry, please. Yes. Yes. You say the critical value multiplied by the standard error. And it made the marginal error. Yes, the margin of error. Then you say this. Right. Okay. So now let's move into the next topic, which is hypothesis testing. So with hypothesis testing, also you need to bear in mind that you can make decisions based on different things. Like when the population standard deviation is known, when it's unknown. And for the proportions. So by the end of the session now, you just need to know certain things relating to hypothesis testing. We're going to just look at the basic principles of hypothesis testing. How you make a decision, how you identify the fact that they are given to you in the question relating to whether you need to do a hypothesis testing for the mean or for the proportions. Like I explained, we will look at three different sections within hypothesis testing. What you need to know with hypothesis testing, it is what the researcher wants to claim or wants to prove. And always it uses the population parameter in your hypothesis statement. We always use the population parameter. So we will use the mean or the proportion all their time, always. And always remember that in your null hypothesis, we always have an equality sign. So it's either they can give it the greater than or equal, less than or equal or equal. Or it doesn't even really matter whether we have those. Always there is an equality sign to it. So it can always just be equal, equal, equal, equal, equal. It will still give you the same. And when we make a decision, we make a decision based on whether we're going to reject the null hypothesis or we're not rejecting the null hypothesis. And when we state the decision, we say we reject the null hypothesis or we do not reject the null hypothesis. There is also an alternative to what the researcher wants to claim, because there are two sides to the coin. You are innocent until proven guilty. It's either you are guilty or not guilty. There is no in between, right? So with hypothesis testing, there is still the same. So you will either have a yes or a no. So your yes will be your hypothesis. Your no will be the alternative of that hypothesis. So with alternative hypothesis, there is no equal sign. So the only signs that go on the alternative hypothesis are greater than, less than or not equal. Very, very important to be able to state the correct side. Because using this sign will tell you whether where is your rejection area? How are you going to reject? How you find your critical value and how you make a decision? It is based on the sign that is located on your or under your alternative hypothesis statement. We don't make any reference to this. We use this information to help us make a decision. But we don't come back and say, when we conclude and come back and talk about the alternative. Only the null hypothesis. And we will look at that just now. There are different types of errors that can happen. You get type one error and type two error. I'm not going to talk too much about that. But when you make a type one error, it is when you reject a true null hypothesis. What do we mean by that? Your null hypothesis is what the researcher wants to prove. But what if the researcher wants to prove a greater than or a less than? Those three cannot be under the null hypothesis. Therefore, it means you're going to be putting them under your alternative hypothesis. And your true null hypothesis will be not the correct one. Right? So that will be your false null hypothesis that you will have there. And when you have that case, you will be creating what we call a type two error. When you fail to reject a false null hypothesis, because your hypothesis would have been in your alternative instead of your null hypothesis. You create a type one error. Sorry, a type two error. A type one error is when you reject if the researcher wants to prove a greater than or equal. And you reject that you're creating a type one error. So most of the time we will be creating type one errors. All right. So always remember the signs corresponding to the weights, because sometimes they will give you weights in state of sight. Like they might say exceeds in excess of you need to know that it's greater than if they say fewer than you need to know that it is less than. So very important. We've learned about this when we were doing study unit six. Oh, what you call discrete probabilities and so on. But remember now with this hypothesis testing also, because we use the signs, you need to be able to reflect in terms of what do the signs mean in weights and in a mathematical symbol. Okay. So this time help you to make a decision. So when we make a decision for hypothesis testing, we make decisions in two different ways, especially for Z test, we have two ways that we can make a decision. We can use the critical value or we can use the PV. When we use the critical value, it's very important as well to know whether are we doing a one-sided test or a two-sided test or a two-tailed test. It is when you have a not equal sign in your alternative hypothesis, that tells you that you are doing a two-tailed test. Remember, I said the null hypothesis, sometimes we will always have an equal sign. So the reason why I'm not mentioning the null hypothesis, I'm mentioning the alternative hypothesis. It is very important to look at the sign that is under there because it tells you whether you're doing a one-tailed test or a two-tailed test. And when it's a two-tailed test, your region of rejections, there will be two side of that. Therefore, it means we divide our alpha by two. We will divide the level of significance, which is our alpha value, by two to find the region of rejection. Anything that falls when we calculate the test statistic, if it falls in the region of rejection in the blue shaded area, we're going to reject the null hypothesis. If it falls in the white area under the cap, we do not reject the null hypothesis. And that's how we will make decisions with regards to the critical field. When it's a one-sided test or a one-tailed test, now depending on the sign on your alternative, you can see that you can state your null hypothesis as equal, but your alternative sign can be less than. When your alternative sign says less than, therefore, we're looking at the lower limit or the lower site of the take. So when we have a set critical value in your lower limit and they ask that the z value is minus 2,33, whether or it's a t value, the critical value. If it falls in the rejection area, we're going to reject. If it falls in the above critical value, we're going to not reject. We do not reject the null hypothesis. The same with the upper tail, when the sign is greater than. If it falls in the area of rejection in the upper tail area, we reject the null hypothesis. Otherwise, we should not reject the null hypothesis. That is one way of making a decision, especially whether we're using a z or a t. When we have the z or where the population standard deviation is known or for the proportion, we can also make a decision by using the p value. So now with the p value, since you will be calculating your z test statistic, which whether it's for the mean or for the proportion, you will use this z value to go find the probability on the table. Remember, we used to do this in the normal probability distribution and also in the sampling distribution where we will find the probability of z less than a value or z greater than a value or z between. So here the z value, which is the z that we know, we're going to use that z test statistic to find the probability. And it's something that you have learned when we were doing the sampling distribution because we're going to be using the sampling distribution formula as well. Now, how you make a decision? If your p value, this is the rule. If your p value, which is the probability you're going to find, probability. Your probability, which is called the p value, if it's less than alpha, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis. So you need to pay attention to this. If your p value is less than alpha, we reject the null hypothesis or we do not reject the null hypothesis. Now, I need to go back one more slide because I didn't explain something clear. Remember, I'm going to explain it here where I said if it's a two-tail test, we divide alpha by two, right? If it's one-tail, we do not divide alpha by two, whether it's less than or it's greater than, we're going to find the critical value by just using alpha. Okay. If you're doing the p value, now there are certain things that we need to be aware of. For finding, let's say we're finding for alpha, sorry, for z of less than, right? If your null hypothesis, let's use the null hypothesis statement, or sorry, your alternative. If your alternative hypothesis, whether it's for the probability or not, if it's less than, right? Then the p value, your p value will be equals to the table value, right? Irgat less of whether your z value is negative or is positive. Irgat less of whether your z value is negative or positive. Remember, your table contains the probability of a less than, right? We always remember that, that your table values contain the probability of a less than. So if our alternative hypothesis has the sign of a less than, when we go find the p value, the value we find on the table, whether on the positive side of the table or on the negative side of the table, that value we find on the table is called the p value. Things change when the following happens. In your alternative hypothesis, your statement says it is greater than. When it's greater than, irgat less of whether you find your z value on the negative or positive, you're going to find your p value. I say one minus the value on the table. Irgat less of whether z is negative or positive. You're going to say one minus the value you find on the table. The last one, if your alternative hypothesis states that it is not equal, which means it is a two-tail test. This is a two-tail test. Whereas these ones, both of them, they are one-tail test. This one is a two-tail test. Now, in order for you to find the p value. So the first one, let's assume that our z value is negative. So when the z value is negative, then your p value will be equals to two times the table value. So you just multiply the value you find on the table by two. If your z value is positive, if your z value, if your answer of your z value is positive, then your p value, oh, sorry, I must not do it equals. Then your p value is equals to two times one minus the table value. So you need to be aware of this when you're making your decision based on the p value. So these are very scenarios are very important. So how you find your p value is very important. Let's look at examples. Oh, no, before we get the next example, there are six steps of hypothesis testing. If you master all six of them, you should be able to answer any questions because every statement in your hypothesis testing or options in the exam or assignment will likely be linked to any of these statements. So the first statement is to state your null hypothesis and alternative hypothesis and remembering always your null hypothesis has an equality sign and we always use the population parameter. Your alternative does not have an equal sign and the sign you put there is very important because it will tell you whether you're doing a two-tailed test or one-tailed test. You need to be able to identify what is the level of significance, what is your sample size, and maybe probably identify what other things you are given, whether you're given the population, your population mean, or you're given your p, so that if you are given x, you are able to calculate your sample proportion. Step number three, you need to determine the type of test you're doing, are you doing a z test or are you doing a t test? It's very important. T test, you do it when the population standard deviation is unknown. Z test, you can do it for the proportion and when the population standard deviation is known. Then step number four, you need to find the critical values. The critical values, if you're going to make a decision based on the critical values, the critical values depends on your alternative hypothesis. If you are doing a two-tailed test, you're going to divide your alpha by two and go find the critical value either on the z table or on the t table. If you are doing a one-tailed test, you're only going to use your alpha value to find your critical value. Then step number five, you need to calculate your test statistic, whether it is a z test statistics for the mean or the t test for the mean or for the proportion. You need to be able to calculate the test statistic and the test statistics, yeah, we are using the z score or the z value formula that we learned in the sampling distribution. It's the same, one and the same. Step number six, you make a decision and conclude. Making a decision can be in two ways. If you are doing the z test, remember you can either use the critical value or you can use the p value. If you are doing a t test, you only use the critical value and then you make your conclusion. With step number six, I always like to draw the diagram so that it makes it easier because remembering all the rules to say if it's a two-sided text, if it's greater than or less than the critical value we reject and so on. I don't have to remember all those rules. The diagram can identify my original rejections and then I make the decisions. Let's look at the examples. I hope. Oh, not an example. I don't know my slide. Okay, so remember what I just said with step number five, calculating the test statistics. To calculate the test statistics when the population standard deviation is known, we use the test statistics and z state and we say the sample mean minus the population mean divided by the standard error. Remember that bottom part is your standard error divided by the standard error which is the population standard deviation divided by the square root of n. For when the population standard deviation is unknown, we use the test the t test which is the t state population sample mean minus the population mean divided by the standard error which is the sample standard deviation divided by the square root of n. For the proportion we use the z test statistics where we use the sample proportion minus the population proportion divided by the standard error which is the square root of your population proportion times one minus the population proportion divided by n. Your p which is your sample proportion if it's not given you can calculate it by using the observation satisfying that divided by the sample standards. Let's look at the examples. I'm not going to look at the answers I'm going to do the hypothesis testing looking at the six steps right and then we will answer the questions. I just want to give you a feel of how we use all six including also using the p value we will do the critical value and we will also do the p value. Okay so for a sample of the five items from the population for which the population standard deviation is sigma equals 20.5 The sample mean is the x bar is equals to 458.0 at the 0.05 level of significance the tweeter wants to test the hypothesis and wants to test the null hypothesis that the mean is equals to 450 against the alternative that states that the mean is not equals to 450 which one of the following statement is incorrect okay so we've got all those statements but what is it that we are given here so now we need to also be very careful when we answer questions and we do get the statements and everything so in hypothesis testing they can either give you a confidence level of 95% confidence level or they can give you the level of significance always know that your level of significance is alpha if they give you 95% confidence interval know that it is your confidence level and you need to find your alpha so what else am I given I'm giving the sample size n I'm also giving the population standard deviation is known in this instance so this one is known so already it gives me an idea and they've given you your x bar they give you your alternative and null hypothesis so statement number one state your null hypothesis and alternative they already stated them I'm not going to repeat them but what I can say here is I have a two sided test because it is not equal so it to sided sided test that's one thing I know step number two state what else are you given I know that my n is d5 my alpha is 0,05 and my population standard deviation is known step number three find what kind of a test is this so since my population standard deviation is known therefore I'm using a z test that's all what I need to state there it is a z test statistic or a z test step number four find the critical value to find the critical value I know that I must use z alpha divide by 2 because it's a two sided test and therefore z of 0,05 divide by 2 which is z of 0,025 0 which I now know that it is 1,96 some of these things you know from the previous one but if it was z alpha I need to know that what is z alpha cannot just assume that it is 1,96 because it's not the state because yeah I know that I'm dividing alpha by 2 which I know that it gives me 0,025 0 and I could go to the table and so I could refer back to the table that I shared with you on the confidence level and the answer will be 1,96 step number five is to calculate our z test statistic which is that state is equals to the sample mean minus the population mean divide by sorry my pen is doing something else divide by the population standard deviation divide by the square root of n and my population standard or my sample standard deviation they gave me it was 0,458 minus the population standard deviation sorry the population mean it's always stated in the hypothesis test which is 0,450 divide by the standard deviation of 20.5 divide by the square root of our n is 35 let's go and calculate so let's all calculate it and see if we get the same answer I'm going to take advantage of my 0,458 minus 0,450 divide by it would be another fraction 20.5 divide by the square root divide by the square root of the 5 is equals to change the thing it's a z test I can leave it to 2 decimal so the answer will be 2,31 the answer here equals to 2,3 that is my answer for my z test so now I can go and make a decision 6 let's make a decision we make a decision by drawing this normally distributed and I know that my critical value is 1,96 so I can just zoom that from here anything that falls here my critical value here of 1,96 and I know this side will be negative because remember at the beginning in the middle for a normal distribution your mean is always equals to 0 so the mean in the middle and on this side we can also create another region of rejection because it's a two tailed or two sided test so that will be 1,96 so anything that falls here we're going to reject the null hypothesis anything that falls here we're going to reject the null hypothesis so that is our region of rejection so anything that will fall on there so now we need to take our Z test statistic and see where does it fall so 2,31 it's positive it will be on the rejection side correct? I can change my channel so 2,31 falls some way the side since it falls there it falls on 2,31 falls in the rejection area and since it falls in the rejection area then falls in the rejection area and we can make a conclusion and say since our Z test of 2,31 is greater than the Z critical value of 1,96 we reject the null hypothesis and conclude that there conclude that there is there I don't know how to write small small prints there my pen doesn't want to write there is sufficient or there is a statistical whatever you want to call it sufficient isn't that the mean is not equal to you is different from 450 that is how you make a decision now this is based on the critical value so now let's go and do when we look at the p-value so we're going to repeat the same so instead of using the critical value let's go and do the same step number 6 using the p-value because we're doing a two tail test so now our p-value will be 2 times now I need to go back and look at the thing remember now my Z value is positive so since it's positive so it will be 2 times 1 minus the table value so it will be 2 times the table value so I need to go back or go to the table value so let's go to the Z critical values Z test and I must go to the positive side and look for 2 comma 3 1 2 comma 3 1 so 2 comma 3 on the left and 1 at the top so when they both meet that's where I need to be so that is 9896 repeat you go to 2 comma 3 on the left and 1 at the top that gives me my 2 comma 3 1 is 9 comma 86 I think 96 comma 9896 9896 9896 0 comma 98 96 minus 1 2 and it will be 2 times we don't even worry about inserting it step by step just go to the calculator because I'm going to run out of space and let's do that is 2 times 1 minus 0.99896 equals bracket equals 0 comma 0208 0 comma 0208 0 comma 0208 that is our p-value therefore we can make a decision, remember the rule is that with the rule what do we know the decision rule the decision rule states that if the p-value is less than alpha we reject the null hypothesis right so our p-value is let me write it here our p-value less than alpha or not less than alpha we reject the null hypothesis that is the decision right that's what the decision rule states so now what is our p-value our p-value is 0 comma 0208 what is our alpha our alpha is 0 comma 05 now 0 comma 0202 and 0 comma 05 it is less therefore we reject the null hypothesis and you can see that both we reject the null hypothesis in both statements and we can conclude that there is sufficient evidence that the null hypothesis is not different from 05 or since we are rejecting that it is required so now let's look at how we answer the question so six steps whether we use critical value whether we use critical value or we use the p-value we still reach the same conclusion which one of the following statement is incorrect number one this is a two-tailed test we did confirm that it is a two-tailed test the critical value is 1.96 we did find that the critical value is 1.96 the test statistics is 2.31 we did find that it is 2.31 the p-value is 0 comma 01 we did find that the p-value is 0 comma 0 to 08 if it was one-sided test it would have been true because 1 minus 0 comma 9 8, 9, 6 is 0 comma 0, 1, 4 so that is the incorrect one and null hypothesis is rejected at 5% level of significance we did reject that at 5% level of significance whether we use the p-value or the critical value we still going to find the correct this or we going to make the correct decision or the right decision like I said you need to know all six steps of hypothesis in order for you to be able to answer questions in your assignment or exam because one of the options or all of the options might be related to all the six steps of hypothesis as you can see from here okay any questions relating to hypothesis testing for the mean when the population standard deviation is known if there are no questions then let's look at another example so here we are looking at when the population standard deviation is unknown how do I know that it is unknown let's read the question a study has done or a study was done in a daily cash balances assume that they are normally distribution of the bank to investigate the hypothesis that the average cash balance do not exceed 25,000 the sample of 100 days yielded a mean of 24,920 and the standard deviation of 300 hypothesis test with an alpha of 0,05 was that the critical value in this case is there is some missing information in my thing above so the critical value in this instance we're going to calculate it because it's not given in this because I think I I didn't take into consideration when I copy from in question 4 probably this was question 5 but we will fix that we have enough information which one of the following statement is incorrect so there is our statement so now you need to pay attention to the statements because we need the 6 steps of hypothesis testing so let's start with the first step of hypothesis testing by looking at what we given so it says the average balance did not exceed 250 that's what the researcher has or the aim of the researcher wants to prove that they did not exceed 25,000 we are also given the sample size which is our N we are told what the mean is which is our X bar we are also from the same sample so if you ask question the sample of 100 you know that the mean and the standard deviation so it means the mean and the standard deviation given here comes from the sample so this is your S standard deviation because it comes from this sample unless if they have explicitly stated there that and the population standard deviation is this but since they didn't say that and the whole sentence reads from the sample so we assume that the standard deviation here is from the sample a hypothesis test of alpha we are given our alpha and we need to choose which one of this is correct so step statement number one do it not exceed let's go to our remember this does not exceed does not exceed means it is less than or equal right so then what the researcher wants to prove cannot be can be placed in your in your hypothesis testing so let's see our null hypothesis states that the mean is less than or equal because does did not exceed or does not exceed it's the same as less than or equal and that will be less than or equals to 25 thousand therefore your alternative hypothesis will state that the mean is greater than 25 thousand that is step number one step number two we need to state what else we are giving our alpha is 0,05 our N is 100 and we are not giving the populations standard deviation it is unknown but those are some of the things that we know please step number three what kind of a test are we doing since the populations standard deviation is unknown therefore it means we're going to be doing a T test we're going to be doing a T test step number four we need to go find the critical value remember how we find the critical value for T test sorry my bad critical value for T test we use we need to also pay attention to the following the sign it says greater than therefore it means that it's one sided test which is one thing that I didn't mention here we're doing a one sided a one sided test so we're going to find the critical value by using alpha and the degrees of freedom and we know our degrees of freedom it's N-1 and therefore our T alpha we were told that it is 0,05 and the degrees of freedom N-1 N is 100 so it will be 100-1 and therefore our critical value will be given by T of 0,05 and 99 so we mean we need to go to the T table so let's go to the T table T table T table there is our T let's go down until we get to the 100 and there is 99 and we're looking for 0,05 so it's the third column 1,2,3 so we can go down to the third column 1,2,3 and 1,6604 is our critical value step number 5 let's do step number 5 we need to calculate our T statistic so we know that T is equals to X bar minus the mean divided by D as over the square root of N now let's substitute our X bar we did identify it in this it's 249 20 minus the population always stated in your hypothesis which is 25,000 divided by your standard deviation as 300 divided by the square root of N of 100 249 20 protect 25 100 0,0,0,0 divide by 300 divide by the square root of 100 and that gives me minus 2.666 so it's 2 decimal remember it's minus 2.67 2.67 okay so let's make a decision our step number 6 we make a decision we draw our region of rejection our critical value we did find it also pay attention the sign says greater than right so greater than will be on this side and we will with 1.604 they because this side is the greater than okay so it's one sided greater than will be on that side okay so now let's make a decision minus 2.6 falls somewhere on this side if in the middle the 0 you know that it will fall in there do not reject so this side is there reject now hypothesis and this side is there do not do not reject the null hypothesis so what is our decision we do not reject the null hypothesis we fail to reject the null hypothesis it's at alpha or 0.05 now let's look at the options we need to choose the incorrect value so choosing the incorrect value let's see statement number 1 it says the alternative hypothesis is less than 250 I just want to double check something in terms of the question that the average balance did not exceed I just want to see our side if we use the right side it does not exceed this is exceed that is does not exceed we did use the right side exceed is greater than therefore does not exceed will be less than so we use the correct because that says did not exceed if it did say it exceeds that it would have been a different question so question number 1 hypothesis you know that it will be greater than because our null hypothesis is less than or equal and this is looks like it is the incorrect what the test statistic is minus 2.7 we use the T table in this because the population standard deviation is unknown so that's what we did there is enough evidence to support the statement at alpha there is because we were able to make a decision we fail to reject the null hypothesis we do not reject the null hypothesis and that is correct so the only statement that is incorrect is option number 1 that's how you will answer the questions just want to see what time is it we are right on top of the hour so I had one last activity and then we are done but before I do that before I do the for the proportions I do have additional exercises that you can also look at so you can take a screenshot of exercise 1 also looking at you are given the population so normally the population is normally distributed with the population with the standard deviation so that standard deviation comes from the population so it's a sigma so you just need to know how to do this in order for you to answer all these questions the second one it's proportions because they gave you percentages and you can also identify by looking at the proportions yeah the other one it's also proportion so let's before you call let's look at this one question consider the following information from question 5 which is the other thing because I copy and paste from the previous tutorial letters so you will find this kind of things in the notes as well most of the schools reported a decline in the number of absences from the education department learner transport and school nutrition in a sample of 200 schools from Joe Babi yeah I'm not sure if I pronounced that correctly district municipality 76% reported a decline in the number of learners absent the district manager is adamant that the true population proportion of schools that reported a decline in the number of absences is different from 78% previously formulate a null hypothesis an alternative hypothesis conduct the hypothesis testing for the true population proportion at 5% level of significance which one of the following statement is incorrect so instead of me doing the six steps I'm just going to answer the questions as I see them because you already know how to answer those questions but first let's identify what we give it so we are told these sample size is n of 200 from Joe with a 76% decline so our P is 76% which is 0,76 and what they want the district manager says there is an the number of absences is different from 78% and because they say it's different so therefore it will be a not equal and equal it's a two sided test that's going to be doing so we're going to be doing a two sided statistics with our proportion which is the now the other thing I forgot to mention those who are doing 1501 remember when I speak about when especially when we get to the proportion our P for you is your P Gabi and our proportion for you is it's our population proportion so don't get confused with the letters you just need to use the correct words for your module as well so you know the six steps I'm not going to go through all the six steps because we run out of time but this is the two sided test since it's the two sided test we should be able to use that information to answer some of these questions so on this question we're looking for the incorrect statement we need to calculate the test statistic that is question number one so we know what the test statistics is that is equals to P minus proportion divided by the square root of population proportion times one minus the population proportion divided by n so substituting the values we know that our P we did find that it was 0,76 0,76 minus 0,78 because that's what the population proportion would have looked like divided by the square root of 0,78 times one minus 0,78 divided by our n of 200 let's quickly go and calculate that 0.76 minus 0.78 divided by the square root of 0.76 0.78 times 1 minus 0.78 divided by 200 and sorry about the noise there are cutting something next up there will be 0,68 minus 0,68 which means this is correct the next one it says what is the P value? remember let's go back to refresh your mind in terms of the P value if it's negative it's just two times the table value, right? for a two-sided test if you forgot just wanted to go and remind you on that so in order for us to find this P value we're going to say two times the value we find on the table and on the table we need to go to the negative side z table using our test statistic so we go to the z table we're looking for minus 0.6 which is that and 8 at the end which is the last second column the last second one which is this one 0,248 0,248 0,248 0,248 which is equals to 4 6 0,4964 which means that was incorrect the alternative hypothesis is that the proportion is not equal yes, our alternative would have been because it's a three-sided test would have sent the population proportion is not equals to 0,78 which is correct the null hypothesis is not rejected, I don't know that let's check the rule says the P value if it's less than alpha we reject the null hypothesis that is the rule so our P value is 0,49 we can do that 0,4964 our alpha we're told it's 0,05 so what is the sign the sign is greater than therefore we do not reject the null hypothesis reject the null hypothesis not rejected that is correct we can conclude that the proportion that reported the decline in the number of absence is not significant different from 0,0 because it's not different if we reject the null hypothesis where our null hypothesis would have stated that the proportion is it's not different the proportion is equals to 0,78 rejected that that it is equal and that is our conclusion for today just to recap we looked at the confidence intervals for the mean where the population standard deviation is known and unknown and for the proportion and always remember that we always use the point estimate plus or minus indicating whether the lower limit of the minus and the upper limit with the plus times the margin of error which is your critical value times your standard error gives you the confidence interval always remember that to find the critical value always remember depending on the table that you are using you divide alpha by 2 by using the level of significance or by using the confidence level coming to hypothesis testing remember the six steps of hypothesis testing where you need to test the null hypothesis and the alternate or state the null hypothesis and the alternative bearing in mind very well that your null hypothesis the statement always has an equality side and it is what the researcher wants to prove and it is what we use to make conclusion also your alternative hypothesis being the statement that is the opposite of your null hypothesis does not have an equal sign and the side you put on there tells you whether you are doing a one tail test or a two tail test if you are doing a one tail test there is one region of rejection so it means when you find the critical value you only use the alpha value when you do a two tail you are going to use alpha divided by 2 now this third step is for you to identify what other information effects you are giving like your alpha value and your N and other related effects within your statement that might help you to answer the question step number three you need to be able to state whether you are doing a T test or a Z test based on the information given whether are you giving the population standard deviation or you are giving the sample standard deviation or you are doing for the proportion and you need to clearly state whether is it a T test or a Z test step number four you need to be able to go find the critical value to find the region of rejection your critical values also remember it's based on your alternative hypothesis testing statement if it is a one sided test you find your critical value by using the alpha value and if it's a T test you are going to use the alpha value and the degrees of freedom which is N minus 1 again you need to be able to calculate your test statistic your test statistic is the same as your sampling distribution Z score based on whether the population standard deviation is known or is for the T test or is for the proportion and always remember for the proportion if you are not given the sample proportion you will be given your X value or your outcome that you can use to calculate your sample proportion the last step is for you to make a decision to make a decision you can either use the critical value and the test statistic to make a decision or you can use your test statistic and your alpha value in order for you to make a decision when you make a decision based on the critical value always remember your regions of rejection if it's a two tail test there will be two sides if your test statistic it's negative does it fall in the negative side of your region of rejection or if it's positive does it fall in the region of rejection for the positive side and you make that decision based on that if it is one sided test depending on which one if it's a less than in your alternative the sign was less than then it will be on the negative side of your region of rejection if it was greater than it will be on the positive side of your rejection area as well you need to be able to do that if you're making use of the p-value also remember to find your p-value if it is one sided test and your value of your z is negative or positive but it is a less than always the p-value will corresponds to the table fail if it is greater than you always going to say one minus the table value in order for you to find your critical sorry your p-value if it is a two sided test remember that if it is negative your z-value is negative then you're going to say two times the table value which is the value you find on the z-span standardize cumulative normal distribution table if it is positive you're going to say two times one minus the table value on the standardize cumulative normal distribution table and that is it are there any questions, comments before I give you your Saturday afternoon off no questions from my side thank you Sissi thank you very much if there are no other questions please remember to complete the register and just go there and stop the recording thank you very much Sissi thank you any