 Good afternoon Everyone I am so sorry For starting late. I hope you are all good Without wasting any time Welcome to your session 16 Where we're looking at confidence interval we continue from where we left off on On Wednesday, do you have any? Comments questions regarding what we did Before we start with the recap Okay, if there are no comments, so let's Do recap let's recap on what we've learned So we learned that we do confidence interval for the population When the population standard deviation is known and today we're going to continue way It is unknown and then when they will do the population proportion What we've learned about the confidence interval for the mean when the population is known that the three assumptions Assumptions needs to be met the population standard deviation has to be given or known The population needs to be normally distributed and if the population is not normally Distributed we need to use the large sample size and in order for us to calculate Or find the confidence interval We use the formula The point estimate plus or minus the critical value times the standard error Which our point estimate for the mean when the population standard deviation is known it's x bar Plus or minus because we have the upper limit and the lower limit of the confidence interval Times the critical value and since is the critical value from from Where the population standard deviation is known so we use the Z Table and we go find the critical value by using the confidence level Which is one minus alpha where alpha is your level of significance times the standard error which is Population standard deviation Divide by the square root of your sample size We also looked at how to we find that critical value to say if we are given a 95% confidence interval we take that 95% confidence interval, which is one minus alpha We find the value of alpha and since because we've got two Boundaries who go and divide alpha by two so that we can get the value of Zero comma zero zero two five and then we take this value of zero comma zero two five We go to the table. We look for the z-value corresponding with this probability And that z-value we find it on the negative side of the table and it corresponds to 1.96 on the negative side of the table and Since we're dealing with confidence interval We just say 1.96 and we have our critical values or our critical boundaries Use those critical values to find the confidence interval We also learned that there are common levels of confidence that we use to test and Especially the 90% 95% and 99% and you can Learn them or you can lend them by heart or You can use the formula and go get the critical value on the table The thing that you need to be aware of is only for 90% you need to always remember that For 90% The critical value is 1 comma six four five Then we also looked at how to find The confidence interval when we're giving information and we use this information this Example where we are given 11 seconds from a large population with the mean resistance of 2.0 and The population standard deviation of 0.35 and remember the three assumptions needs to be stated so We are told We use a normal district a normal population so Assumption number one and assumption assumption number two and three I met because it says the population needs to be normally distributed and if it's not normally distributed Then we use a large sample size a large pop elapsed sample size. So yeah, the population is normal so We are also given another exam assumption is you need to Check if you are given there Stand population standard deviation. We are given the population standard deviation which is also called the sigma And Calculate the value by using point estimate plus or minus the critical value times the standard error We know the critical value for a 95% since the question said at the 95% confidence interval Go to the table we go find the one point nine six and Which is the nine point once one point nine six that we already found and then we substitute it into the formula standard population standard deviation divided by the square root of the sample size which is 11 and found that the two boundaries or the mean lies between 1.9932 and 2.4 and we said also we can write it as 1.9932 I will call on Let me call on 2.40 6 8 and that's what we've learned Last week and we also learned how to interpret it, but it's not Required for you to know how to do the interpretation so today we're going to continue and We go into continue learning about the basics some of the basic concepts of Confidence interval we know that we've learned how to construct the confidence interval where the populations that are deviation is known So today we're going to learn where the populations standard deviation is Unknown and then on Wednesday we do the proportion okay so when We construct a confidence interval where the populations standard deviation is unknown. We go into a zoom that If the population standard deviation is not given to us then they would have given us the sample standard Deviation which is the s So they will give us the sample standard deviation. You need to pay very Close attention or be very careful when you read the question Sometimes the question might just straightforward and say the sample standard deviation is or s is equals to or They might say from a sample of This much The standard deviation is this much from a sample of This much, so let's say our sample size is 11 from a sample size of 11 the standard deviation found was 3.5 I Just reading the question like that You clearly can see that the standard deviation. They are talking about is Coming from the sample So when you read the question you need to look out for more cues or keywords Things that will tell you whether are you working with the population standard deviation? Or are you working with the sample standard deviation and Since because we're working with the sample standard deviation This will introduce some uncertainties in terms of the things that we are doing because The sample sizes or the samples that come from multiple samples differ from one sample to the other and in that case Since we're using the samples standard deviation We then going to use the T distribution In state of using the normal distribution Okay, or as remember also when we find the confidence intervals The same will still apply Going to use the formula Point estimate plus or minus the critical value times the standard error now our Critical value will not come from the cumulative standardized normal distribution distribution Z Value it will come from the T distribution table Your standard error will still stay the same because it will still be your standard deviation Because now it will be the sample standard deviation divided by the square root of your sample size Your point estimate for the mean will still remain as your statistic or Your X bar or your sample mean and the formula Will look like that X bar plus or minus the critical value, which is T alpha divided by two So therefore it means we still going to have to find the level of confidence and use the level the And use the significance level or the level of significance alpha to go find the critical value on the table But now the T table looks different. Yeah, I'm only having T alpha divided by two The critical value when we get to it We're going to find it by using T alpha divided by two and the degrees of freedom And our degrees of freedom is n minus one But also similarly with confidence interval for the mean way There Population standard deviation is unknown There are a couple of assumptions that needs to be met in order for you to know that you're going to use the T distribution One the population standard deviation will be unknown Or it will not be given to you The population needs to be normally distributed and if it is not normally distributed Then we need to have a larger sample size Also for T distribution So the only key thing here which differentiate between what we did on Wednesday and what we're going to be doing today Is assumption number one So let's learn how to find this Critical value to find the critical value. We're going to use the T distribution table and Because we use the T distribution table like I said Our critical value. We find it using T alpha divided by two and the degrees of freedom We're going to look at that just now What's the distribution as the value of your critical values increases? They tend especially when n increases The values of your T distribution Tends to become Your normal distribution and we will look at the table just now As the value of your n Which you use to calculate your degrees of freedom The value as the values of n increases The T distribution tends to become normally distributed Let's look at how we find the critical value So we're going to go to the actual table, but for now I just want to use this slide to demonstrate So we know that our table Oh Of the z distribution table We have our z values at the top and the last digit at the At the top Sorry on the left and then other values at the top and we have the The probabilities within the table and we use this to go find the critical value With the T table It works different Your table is split in terms of your degrees of freedoms and your Alpha divided by two at the top We're going to go to the table just now So we always going to be using the upper tail areas of the table On the left you will have your degrees of freedom And at the top you will have your alpha divided by two values So let's look at an example here where we have we are given At 95 percent confidence interval with sorry at 90 percent confidence interval So this is at 90 percent confidence interval and n is equal to three Find the critical value, which is t alpha divided by two And your degrees of freedom Which is t alpha divided by two And n minus one To find that you go into find our alpha first Our alpha remember it is One minus alpha is equals to zero comma nine which Our alpha will be one minus zero comma nine Which is equals to zero comma one zero Which is our zero comma one zero there our alpha of zero comma One zero will be zero comma one zero divided by two and n is three minus one Therefore our t alpha will be zero comma zero comma zero five And two so we're going to use The two values our degrees of freedom of two And alpha zero comma zero two so we go to the top of the table And look for zero comma zero five And go to the left side of the table and look for two where they both will meet That's where we will find our critical value Let's do another exercise example. We'll do it on the table itself We'll do more Exercises on the table We need to go to the t table and usually I think it is immediately after the normal distribution table So you can find your t tables in your book in your study guide in at the back of your past exam papers Oh at the back of the prescribed book You need to make sure that you use the right t table Okay, this is the t table We're going to use Right now it's called table e three And it is also called the critical values of t When you use this table, you can ignore the top part where it says cumulative normal cumulative probabilities That top part we will never use You can just ignore that If you look at the table, you will see that your degrees of freedom are on the left and they run to The table goes over multiple pages as well And remember when I said when the values of t increases of n sorry when the values of n increases especially when it comes to the degrees of freedom the values of n increases you will notice that They become Normally distributed. So this is the same as 1.28. This is 1.6 four five This is 1.96 And this is 2.3 3 3 and this will be 2.58 and this are your normal your normal distribution Critical values as you can see The first one is 10 percent. So for 10 percent for 95 percent Remember 95 percent. It's 1.96 So you will see there is 0.025 for that 95 percent confidence interval And it's the same as that and that is what I was explaining But anyway, we were not looking for those ones. Let's do one activity We can use the top part so fine at a 95 percent confidence interval where n is equals to 12 Find the critical value So to find that critical value We say Our critical value will be alpha divided by 2 And the degrees of freedom which our degrees of freedom Is alpha divided by 2 Which is n minus 1 And therefore Because we know what at 95 percent is will be 0.95 Is equals to 1 minus 0.95 Uh, sorry is equals to 1 minus alpha Which then alpha is equals to 0.05 So then we can come here and say t 0.05 Divide by 2 And our degrees of freedom n is 12 So that will be 12 minus 1 Which then gives us t t of 0.025 0 And the degrees of freedom Of 11 So it means we need to come to the degrees of freedom and look for 11 And go to the top and look for 0.025 On where it says upper tail areas and look for 0.025 Where they both meet that will be our critical value And that is our critical value I want to I'm going to give you two minutes Or three minutes Find the critical value where n Is 6 At a 99 Percent Confidence level Confidence interval Do we have the answer I gave you ample of time now Let's see 4.032 that's what Lady say Let's see So To find alpha we say 0.99 is equals to 1 minus alpha Which means alpha will be equals to 0.01 Which is 1 minus 0.99 So to find t alpha divided by 2 and n minus 1 T will be 0.01 divided by 2 And our n is 6 minus 1 Therefore our t Will be 0.05 And the degrees of freedom off So come look for degrees of freedom of 5 and look for 0.05 at the top Where they both meet Should it not be 0.005 Yeah, why did you guys leave me until I get to the answer? 0.05 and 5 because I didn't even calculate that okay So then it is The last one Which is 0 comma Which is 4 comma 0322 Yeah 4 comma 0322 So do you get the Do Do you get it? Do you know how to do to find the critical value on the t table? So let's go do an example So that we can We can find more activities That we can use I'm not going to go too much into this slide because that's what we just explained right now Let's look at that example A random sample of n Equals 25 That's the mean of 50 The standard deviation of 8 Form a 95 confidence interval They haven't given me Sigma so it means my population standard deviation is unknown They have given me s which is my sample standard deviation Therefore it means I'm going to use the formula x bar plus or minus My critical value of t alpha divided by 2 and the degrees of freedom Times the standard error which will be my sample standard deviation divided by the square root of n Substituting everything that is given I'm giving 50 plus or minus My critical value I must do it outside so that I can just only substitute the critical value So my t alpha divided by 2 and n minus 1 Will be given by I'm told I must do 95 confidence interval 1 minus alpha of 0 comma 95 Therefore alpha will be equals to 0 comma 0 5 so it means my t will be 0 comma 0 5 divided by 2 And my degrees of freedom I'm told that n is 25 So it will be 25 minus 1 And this will be t of 0 comma 0 2 5 0 And 24 Then I must go to the table on the table. I must look for t 0 comma 0 2 5 And the degrees of freedom of 24 So t 0 comma 0 2 5 It's this column My t of 24 my degrees of freedom Of 24 t 0 comma 0 2 5 and the degrees of freedom of 24 there is 24 where they both meet That's my critical My critical value is 2206 39 So 2 comma 0 6 39 is my critical value Then I must multiply this. Oh, sorry Multiply By the critic by the standard error, which is my sample Standard deviation of 8 divided by my square root of Sample size, which is 25 We need to calculate the other side So 8 divided 8 divided by the square root Of 25 which is 8 divided by 5 I get 1 comma 2.0639 Times 1 comma 6 The side is 50 plus or minus And if I multiply The answer I get by of 1 comma 6 Multiply by 2 0 6 39 I get 3 0.3 0 22 and remember We can then split At this point we're going to create two Two values. So we create the upper side, which will be 50 minus 3 0.3022 And the upper side, which will be 50 plus 3 0.022 And therefore This side we get 46 4 and 6 9 8 And on the other side we will get 53 4 and 022 Which is the same as The exercise we have Yeah Any question if there are no questions Yeah, it's your exercise. We've got 10 minutes Um, if you are done before that time, please let me know if you are done Um, and then within five minutes or so after five minutes I will ask you if you need extra time But I just need an indication to know if many people are done Also, you can use the chat To post your answers As well. It will give me an indication Okay Not getting any of the options. I'm going with option five reluctant. Please fees or says option five Okay Let's see What are we given we are given the sample mean of 900 the standard deviation Of 30 Sample standard deviation, which means the population is not known We're giving s And our sample size so let's Substitute into our formula plus or minus our critical value Times our standard error Let's go find our 0.95 One minus alpha which will be alpha will be zero comma zero five Our critical value of alpha divided by two at this point I am not even going to Go with substituting alpha of zero comma zero five divided by We know what that is because we know what alpha divided by two is zero comma Zero two five, so I'm just going to substitute t of zero comma zero two five And we know our n n is Hundred, so if our n is equals to hundred Therefore our degrees of freedom will be hundred minus one Which is 99 So let's quickly just substitute into this formula we will go and find the critical value the mean It's nine hundred Plus or minus. Let's go find the critical value on the table. We are looking for 99 And zero comma two five. Let's first find where zero comma two five is in the third column So zero comma two five and 99 we need to go to the bottom And look for 99 85 86 90 91 92 93 97 98 99 and we're looking for the third column one two three third column from there Right, which is this column zero comma zero two five And the degrees of freedom Of 99 Which gives us one comma nine eight four two so we find One comma nine eight four two One comma nine eight Four two And we substitute times our standard deviation 30 divided by the square root of our sample size Which is hundred And we calculate the values so We have 30 divided by the square root Of hundred equals three Multiply that with one comma nine eight four two nine hundred plus or minus five comma nine five two six, so let's expand it nine hundred minus five comma nine five two six and nine hundred plus five comma nine five two six Are you also getting the same? Yes, I'm getting the same minus five comma nine five two six equals eight nine four comma zero four Seven four, so probably there was a narrator on this question as well I don't know I am Just want to double check the values quickly Maybe I copied the wrong question No, copied the right question We getting eight nine four point zero four seven four On the other side It would have been a plus five point five two six I don't know So there was an error on that question so we can ignore all All the values that are there We can ignore all these values and use the ones that we get So we have nine zero five point nine five two six And on this slide We have nine hundred minus five point nine five two six four point zero four seven seven four Which is none of those values that are reflected here Even not even close Let's look at the second one With the second one it's also the same but we need to use Our critical value will be t alpha divided by two and the degrees of freedom will be t At 99 percent alpha is zero comma zero one And at alpha divided by two it will be zero comma zero zero five So therefore the year will be zero comma zero zero five And our degrees of freedom on this one will be Uh, the same as what we had previously it will be hundred minus one which is 99 So we go to the table We go to the table I don't know. I should have shared my entire screen This of moving between the screens back and forth. It's not wicked Just give me a sec. Let me share my entire screen is just that my screen has so many Things opened right now So we need to go to the table the t table We're looking for 99 And zero comma let's go find where zero comma zero five is at is the last is the last column And we're looking at the last column at 99 Sorry to the bottom We go At 99 it's two Please call it out for me. Sorry Let's go write it 6264 So our critical value is two comma 6264 I forgot now Here's 6264 6264 Let's take it up x bar plus or minus the critical value With the use of freedom as divided by the square root of n Our x bar is 900 Plus or minus our critical value. We did go find it. It's Two cover 6264 6264 Times our standard deviation of 30 divide by square root of 100 Which take me calculate an r We did calculate this site and We did find that 30 divide by square root of 100 is three multiply that With 2.6264 We get 900 plus or minus 7.8 792 and we expand it 900 minus 7.8 792 And 900 plus 7.8 792 So if I do the plus first I get 907 point 8 792 And if I do the negative 9 2 point 1208 So is the answer like that? Also not on this options Because that one is 862 this one is 864 852869 I think when they did the options on this questions Either they did the wrong They did the Z value since date of the T distribution Okay, and Let's go back to the previous one Because the next question is linked to both of these questions So remember this one we had the answer as 89 894 To 900 5 the second one we have 892 and 900 and So by looking at the two remember last week On wednesday we discussed Someone's mic is on and they are moving paper. Okay So last week we discussed the two remember that we found The answer for the two because we looked at when the sample size increase What happens remember that that we said when the sample size increase the The the the graph becomes narrower And when it decreases The graph become wider Because we divide the standard deviation by the sample size So when this size is smaller it doesn't affect too much of the other side But when it's bigger Or when it's smaller Then the standard deviation becomes bigger and it affects the other side So we covered that on wednesday So today we can look at the two because now we're using the same information We changed the confidence levels So now let's go back to our previous questions Oh before we go to the previous question So this one says when the confidence level increases So what it says if I have A 90 and 99 99 is increased from 90 to 99 It says if it increases so the bigger the confidence level The wider your Your your boundaries So it says so if this is the bigger the confidence if this is minus three And this is minus and this is positive 10 Then it means for a decrease it says it's narrower So for the decrease If this question says Then It will be like there. So this will be eight and this will be minus one The big the bigger the confidence interval the bigger the Confidence levels the wider the confidence levels or the confidence intervals The smaller the confidence level the Smaller the The confidence interval. So let's look let's go and test that assumption So here we have 95 here. We have 99 So 99 has 892 So 99 has 892 and 907 that is 99 If we go to 95, which is smaller has 894 and 905 uh 894 905 894 Will be somewhere here for some reason. I cannot get my menu for the Oh It's hidden My menu for the pants as you see now The reason why I don't want to You're gonna see things that not supposed to see and I'm gonna share things that I'm not supposed to share Because I'm sharing my entire screen Oh, come on. Come on. Sorry. Sorry. Sorry. Sorry. Let me and let me and share my entire screen It's going to be very frustrating We'll have to deal with that because I don't want to share things that I That I don't want to appear on the video okay, so Let's come back here So at 95 It's 894 and 905. So let's go back there 894 so Actually there was going to the wrong one 894 And 905 So This is 95 percent and This is 99 percent So with 99 percent it says if it's bigger It is Wider so the bigger the wider Narrower So let's go to our question now The question says so remember We had Did we say narrower or wider wider bigger is wide And Small it's narrow So if we come here and answer the question When we only Use the confidence level it increases when the confidence level increases The estimate becomes wider So that is correct And the second one says when the confidence level decreases the The Confidence estimate becomes narrower. So that is correct So the only thing that we can choose from is option number one and option number two to see if they are correct or incorrect I'm not sure if we did answer that on wednesday. So but we can do that We can take one of this Questions that we have already What we can do Instead of using a hundred Let's say this our n but then it means also our It will mean our our Will change also. So all of them will change And it will also mean if we reduce let's say our n reduced to 50 Then it means we need to recalculate this in order for us to test the The logic So this is meant only to answer question the question number three So we already did the other part So You need to go and find the degrees of freedom For t zero comma zero two five because we well, which one are we using now? 99 so we're going to go away. It is 99 Which is at zero comma zero zero five And our degrees of freedom of 49 What do you get as a critical value? I'm not going to the table What is the degrees of freedom? 2.680 2.8 Eight two point six eight zero two point six eight zero Zero Okay, so let's calculate What we have we have 30 divided by the square root of 50 Equals four comma two four two six Multiply by 2.680 Gives us 900 plus or minus 2.6 900 Plus or minus 11 point three seven zero three This will be 900 minus 11 point three zero Three seven zero three and 900 plus 11 point three zero three So let's quickly find that So minus 900 minus 11 point We get eight eight 8.63 I'm just going to keep two decimals And on the other side Where we have 900 plus We have 911 point 37 so now Remember We're talking about the N now right Because now you still have the answer for this one. It was 90 What was the confidence level for the original one? eight nine two point one two two nine zero seven point eight four And this side was 907 Point whatever the number is so now Based on the two So if we look at And that's what it says So if we look at The sample size of hundred we got eight nine Nine two and if we look On 907 this is For n equals to 100 So let's look for any Less so if we have n of 50 we get eight eight eight eight Eight so it will be here And nine 11 it will be there So this is for n equals to Equals to 50 So now Let's make a decision When n is small Or when it decreases It is wider When it is big Then it is Narrower can you see that? So let's go to our question Small wider big narrow. So it's the opposite of what we had so this is for confidence level For n when it's big It's narrow When it's small So small n n n when it's small is wider So let's go only when the sample size increases So when it becomes bigger Says it's narrow as That is correct And when it decreases when it's smaller It becomes wider Which is then it is correct. So it means this and this and this and this are correct So which statement is incorrect? None of the above Is the only option that you can choose Because all of them are correct None of the above meaning none of this statement is incorrect And that's how you go into answer some of the questions By calculating some of the values and making sense of them in order for you to be able to So let's move on to the next one Okay Moving to the next one Reading this question it says The human resource director of a large corporation Wishes to study Absentism among clerical workers at the corporation's central office during the previous year A random sample of 25 clerical workers reveal a mean 27 days With the variance of 16 days Assuming that the population of absences is normally distributed The 95% confidence interval for the average number of days Of absence for clerical workers last year is Can you see how they posed the question? A random sample of clerical workers Reveal a mean of 9.7 days With a variance of 16 days Because the variance is on the same sentence With the sample we can assume that this variance we're talking about this is the Sample variance, which is s-squared We need to go find our s By just taking the square root of your x-squared So we take the square root of 16, which is equals to 4 And since the population standard deviation is also not given so we're going to use x bar Plus or minus the critical value alpha divided by 2 and the degrees of freedom Times s divided by this square root of n The square root of n So we need to go find the critical value t alpha divided by 2 n minus 1 Our alpha They have given us 95% By now you should know that your alpha at 95% is 0 comma 0 5 And alpha divided by 2 is 0 comma 0 2 5 Our n they have given you is 25 So it's 25 minus 1 therefore t 0 comma 0 2 5 and 24 go find the Critical value And tell me What the value is If you are still lost and we don't know how to find the critical value Please talk now Ask Because i'm not long i'm going to look for it I'm going to wait for you to give me the critical value all the time 2.0 6 0 Okay, what does our mean Is it not 6 4? Oh, yes, you're right. It's 2.0 6 4 I need to use a ruler 2.0 6 4 what is our mean? 9.7 9.7 Plus or minus our critical value we did go find it 2.0 6 4 Times our standard error sample is 4 Divide by the square root of n our n is 25 Do the calculation Let's do the calculations So We have 4 divided by the square root Of 25 which is 4 divided by 5 is 0 comma 8 Multiply that with 2.0 6 4 we get 1 comma 6 5 1 2 1 comma 6 5 1 2 did you get it the same as mine? And we expand to 9.7 minus 1.65 1 2 Oh no close the bucket and we go 9.7 1 0.65 1 2 let's Calculate minus 9.7 minus 1.6512 we get 8 point The answer is sometimes left at 4 decimals 3 decimals. We're going to leave ours at 4 decimals as well 8.0 for 8 8 for now and then we can look at which options later on And 9.7 plus 1.6 We get 11 point Maybe I must use points not commas because it complicates my work This is point point 3 5 1 2 do you get the same? So if we're looking let's start with the process of eliminating So Almost most of them they might be correct all of them So like for example This I will just look at the last digit and make an And say maybe or maybe not because they Almost the same. So if I look at this Also, the last bit is different to what we have So I'm just gonna leave it there and to the 3 If I run it to 3 decimals, this won't be right If I leave it to 3 decimals as well that won't be right because I have 11 they have 15 they have 16 So yeah, I have 11 11 So we just need to check the following So let's run off to 2 decimals Our answers Because then we have the 2 to compare 2 So 8 point 0 5 if we run off correctly And the site will be 11 point 35 So This won't be right And the answer will be option number 2 um This is very tricky because also this option In a way it could have And you might you you might you might say oh, but that looks almost closer close to But they have the correct answer there. So when they run it off So We have 10 minutes And I'm not sure in that 10 minutes whether we We will be able to oh, this is the repeat of the same question We untouched so we don't we don't even have to look at this Let's see are there more questions. So there is another question you can do this. Let's see um We have 10 minutes and I think you can Still do this Uh from the information given below At 90 percent confidence interval. So it means We're going to use alpha of 0 comma 1 0 Our x bar is 200 our n is 100 and our s which is sample standard deviation is 5 So we're going to use the formula plus or minus Alpha divided by 2 and the degrees of freedom s divided by the square root of n and we go find t alpha divided by 2 and the degrees of freedom So we know that that is t is 0 comma 1 0 n is 100 minus 1 I'm sorry this was divided by 2 so So Our t will be 0 comma 0 5 n 99 You must go find the critical value 1.660 1.660 So we substitute our x bar Is 200 Plus or minus our critical value of 1.660 Times standard error, which is 5 divided by the square root of n She's 100 So do the calculations And give me the answer for 5 divided by the square root of 100 times the standard error The critical value 5 divided by the square root of 100 Multiply that with 1.660 What do you get? I get 0 comma 8300 Do you also get the same? Yes So then we have 200 minus 0.8300 And 200 Plus 0.8300 200 minus 0.8300 we get 1 9 9 Point How many decimals do we leave this? 2 The answer is in 4 decimals 4 decimals 3 decimals 3 decimals And integer And I already can see that I don't get the answer that they gave me As options Also And then on the other side it is 200 plus 0.83 Which is 200.83 So I'm going to assume that And that are close That is not close. That is not close. This is even way out That is close But it is small because it's 1.8 If I round it up to 2 decimals this This will be 1.7 which will almost exactly be the same as what we have This is 8.2, which is way out. This will be 8.3. So Option number two Unless it's my calculator that cuts off some decimals Let's see again 5 divided by the square root of 100 Is 0.5 Multiply that by 1.60 6 0 0.83 Okay And with that we are Done for the day Just to summarize and recap on what we just did today We looked at some of the basic concepts again of confidence intervals We learned how to construct the confidence interval for the population mean when the population standard deviation is known And with that it concludes today's session If there is any comment Question query Clarity any The platform is yours If there is nothing Then enjoy the rest of your Day and your weekend I will see you on wednesday When we look at Confidence intervals for the proportion We're going to go back to using the z table I hope you all submitted your assignment to you and you are busy with your assignment three Because the content that we're working on now Is for assignment four And I know that your assignment three assignment one two until three. I think they were extended And your assignment three is extended until the fifth Of july So please check your my life email as well to To check all the notification that are sent by your lecture As well If there are no questions Then we can call it the quits and go have fun and enjoy our weekend Bye everyone Thank you for coming I'm busy