 Let's give it some time. It's still starting. Welcome to another session where we do the last bit. The time. We were busy with confidence intervals when they we stopped the session yesterday. So we're continuing. So we already did two questions from confidence intervals. So let's look at this second question. The average cost per night of a hotel in Port Elizabeth. Township is 273. Assume this estimate is based on the sample of 45 and that the sample standard deviation is 65 for a 95 percent confidence interval. What is the critical value? What is it that they have given us? Let's unpack this question so that it helps us to find where are we going to find the critical value. So let's make it bigger. So what is it that they gave us? They gave us the average, the average, which is the mean which is 273 but we don't need that. What else did they give us? The sample size, right? Did they give us sigma or did they give us s? They gave us s. Pardon? They gave us s. They gave us s of 65 and we need to find the confidence interval or the critical value and they gave us the confidence interval 1 minus alpha of 0.95. Therefore it means they have given us alpha of 0.05. So when do we find the critical value? If the population standard deviation is not given or known. We find it on the T so we'll find it by using T alpha and the degrees of freedom which is n minus 1. Our T alpha is 0 comma 0 5 divided by 2 and the degrees of freedom though n is 45 minus 1. It will be T of what is 0 comma 0 5 divided by 2 will be 0 comma 0.25 and 45 minus 1 is 44. So we need to go find the critical value on the T table. Scroll to the T table and this is our T table. Now remember on the T table we do not use the top part, right? All these values at the top we don't use. We use the upper tail area and the values of the upper tail area closer to the data values. So our degrees of freedom is 45 and we're looking for 0 comma 0 25. So it is 1 2 3 or 1 2 3 4. The one in 4 if you come from the right or from the left if you come from the right it will be number 4. So 45 is here 1 2 3 4. I think it's 44 not 45. Ah yes you are right it's 44. At least you are awake. The answer is 2.0154 Let's see the answer is 2.0154 So the answer will be question number or option number B and double check so that we don't make a mistake. 2.0154 2.0154 If I chose the wrong one I would have chosen this because I went to 45 C Are there questions before we move? Because remember if I move I can't come back. No questions. Now we're moving on to hypothesis testing In testing the hypothesis the null hypothesis says the population mean is equals to 15 and the alternative says the population mean is not equals to 15 So you should already hear tell yourself what is it that we're doing is it a one tail test or a two tail test. It's very important to define it based on the hypothesis testing statements. The following information was given so we are also given some information Population standard deviation is 5. The sample mean is 15.1 and the sample size is 10 The level of significance is 0.03 Suppose that the test statistic calculated is 1.96. What will be the p-value? So they gave you so much information that they want to confuse you. Don't get confused. Let's look at it. Let's dissect it. The first thing we can look at is the hypothesis test statement. We are doing a two tail two tail test. That is what they told us based on the sign on the alternative hypothesis The question is asking us we need to find the p-value In order for us to find the p-value we need to find the z test statistic. They told you that the test statistic is 1.96 1.96 Now what you need to remember with two tail test ignore what the level of significance is and the sample size The thing that is important is that they told you that they gave you the population standard deviation therefore it means this is a z test because we can only go and find the p-value on the z table. Now there are a couple of things you need to remember. If your z test is positive, let's not make it positive there If the answer of your z test is positive and you're doing a two tail test then your p-value will be equals to 2 times 1 minus the table value That is one thing that you need to remember. If your z-value is negative then your p-value will be equals to 2 times the table value If you don't want to use this type of scenario, maybe I should just scroll it to the side. If you don't want to use this type of a scenario you can say or let's change the pen. I like it and let's use the other color which I don't think you can even see when I write but you can say it is 2 times not 2 times you will say 1 minus the table value plus 1 minus the table value If you want to use this scenario you will say the table value plus the table value That is one way of finding the p-value. Let's go back to our question. Our z test is positive. It means we need to go to the z table and we will go to the positive side of the z table What are we going to do when we find our p-value? The first scenario because our test statistic is positive so let's go to the z, it's positive. We are going to find our z-value of 1.9 and 6 and we are going to find the table value which is 0.97 50 In order for us to find the p-value we will say 2 times 1 minus 0.97 50 That's what we got 50 What is your p-value? 0.05 0.05 will be your p-value Let's see if you are right 0.05 is our p-value Are there any questions? Let me also explain for the last time this for a two-tailed test this is the scenario. This is what you are going to find or how you are going to find your p-value If your z-value of the z-test statistic, if it's positive p-value will be 2 times 1 minus the table value because the table value has bigger probabilities If you are going to use the bigger probability and multiply it by 2 you will get a probability of more than 1 and we know that probabilities can never be bigger than 1 and they can never be smaller than 1 so you cannot have a negative probability or a probability of bigger than 1 If it's negative, we have smaller probability so we just add both of the two areas that is for a two-tailed. If it's a one-tailed this is the other thing that you need to remember If it's a one-tailed and the sign says less than on your alternative, this one will say it's not equal in your alternative. We always are interested in the h1 value of the sign If you have a z, a one-tailed, where it is less than to find the p-value your p-value will just be the table value That's easy What about when it is one-tailed but it is greater than? In that instance, when it is greater than then your p-value will be one minus the table value Can you see that the same scenario that we use in the probabilities, it works also on the hypothesis, so you just need to remember all this So whatever you have learned in the basic probabilities also in the normal probabilities then to use the same concept when you do your hypothesis testing So always remember that Let's move on to the next question unless there is a question Please don't say we want to when I'm done moving You want to ask a question, so let's go to question 20 Oh at least the picture is the question 20 We need to make it bigger so that you can see To test the hypothesis the null hypothesis says the mean is equals to zero and the alternative states that the mean is greater than zero So we already can tell from the hypothesis that we are doing a one-tailed test based on the alternative side A statistics practitioner randomly sampled 100 observations and found that the mean is 106 which is h bar and the standard deviation s, so this s already tells you what standard deviation is this population or the sample So if it's a sample, therefore it means we're going to be doing A Is it a Z or a T test It will be a T test So some of this information you just need to make sure that you understand what are they and what are they feeding to because they will help you answer some questions Assume a significance level of 1% to calculate the test statistic So what they are asking you to do is to calculate the test statistic Let's go back to our black pad T stat is equals to your sample mean minus the population mean divided by your sample standard deviation So if you have this formula written some way in your hypothesis testing where the population standard deviation is unknown you will use this So let's substitute the values Our x bar which is the mean for the population or for the sample sorry it's 106 The population mean is given in our hypothesis statement divided by your sample standard deviation of 35 divided by this square root of 100 So do the calculation So do you mind on the screen Have you ever invented this calculator, the Casio calculator I respect that they say they have a lot of lives Before I give you my answer I got number D Number D which is 1 comma I didn't get D E 7 for 1 for 3 So in the exam as well you need to pay close attention to the numbers because all of them look exactly the same Can you see it's just the other one has 4 the other one 7 is before the decimal this other one the values are swapped around which makes only E correct So let's see if E is correct Based on the answers we move on to the next question This is one of those Now the responses are all hidden Yeah that's what happened earlier to me when I did the attempt Okay let's fix that I'm gonna have to close Stop sharing so that I can see what is doing this Just have to edit the question because I'm not as a student today I'm not even as myself Let's see I understand it because I can see them I cannot say let's use them the way they are Of course there are randomized who will end up choosing the wrong one I know which one is correct It's not allowing I bet Okay I did send the screenshot yesterday right via whatsapp yesterday before yesterday I think we can use those screenshots because I'm not winning Yeah go too much Can you just please I don't know why it's doing this I still can't get it to work, only the header is working Even if I change the questions I have two of them are working now I'm not all of them I'll open the questions from We'll use the pictures Now I can't find those screenshot When did I send them? On Sunday Yes on Sunday Yes on Saturday Yes on quarter past three Almost all of them I'll just download them and then open them as pictures I can't find them on my phone if you'd like I don't know if it will work though Other people might not be able to see And it's fine I have all of them now Let's go back there What did I do now Let's go back there What did I do now Because this will not work There's only two shows and the rest don't I don't know why I know which one is correct So we will choose the right table based on the responses that we have So let me open the questions This is the question that we are dealing with The trustees of a company pension plan has solicited opinions of the sample company employees about the proposed revision of the plan A breakdown of the responses is shown in the table below We want to test if there is enough evidence to infer that the responses differ among the three groups of employees and this is our choice Which option provides the correct expected value So remember expected value The formula to calculate the expected value is your row total multiplied by the column total divided by the grant total That is one way of calculating You don't have to do the entire table You can just do for one or for two of the values And then come and look at the answers And look at the responses Because you can see that the expected values are different than every one of them Alternatively you can go to the template But you will need to understand which one is that It's a two by three table So you will have to choose A three I can't get my So we want A three by two contingency table Which is this table So if we complete the values on this table So we will just remove those values And put there 67 Even though it's not those I can change them as well And white color Many just And this will be four And against then we can just add the values 82 and 11 63 and 18 And 9 Outside and they are Your expected values and then you can choose whichever one is correct So we can see the expected value that is correct Is the table with 71.5 58 And the rest of the others are different So option A on this question Will be correct So I will have to find time to do this manually now So let's go back to our question Since only the question with the option that is correct Is visible it makes it easy for us to choose which one We know that that was the question The response And that is correct So you can use the template Alternatively you can come and calculate The contingency table So it means you will have to calculate the total So 67 plus 63 For blue color Total times If I'm calculating for For four So the total for For total And the grand total Based on our table because I'm lazy It's 180, 50, 20 So it's 130 So it's 130 And across The total is 110 So 110 Divide by The grand total is 200 Divide by 200 So if we take 130 And multiply by 110 Equals divide by 200 We get 71.5 I can just Check all the answers I can see that Only option A Is 71.5 And that will be your correct answer And you move on That is in the exam Don't try and answer all the other questions Unless you have 75 twice Then you move to the next one But the sooner you get to the answer You move on So let's go to The next question I hope with this one It works And it also doesn't work But we do have The question So we'll put it up there It is The Excel It's not It's not Let's give me a second I just need to find The screenshot of that one It should be An Excel output They didn't download All of them Just give me a second We'll go back To the From to Alta E2 I'll use this screenshot Consider there Following Excel output Testing For independence of two Variables So they have Calculated This is an Excel output It has the rows, number of two rows And three columns And they calculated the photos Then they also went on The statistics How many degrees of freedom Are there and the p-value And the critical Value So they've got all the values All what you need to do is Answer the questions I'm just going to make it bigger Than what it is right now I hope you are able to see If not, we can Revert back to The question on The actual The way I want them to Work We have the data I hope you are able to see Even if it's small So let's see if we can answer this question So number A it says Since 0.31 which is Our test statistic Is less than our critical value Which is 5.992 Then the null hypothesis Of the independence cannot be Rejected, so what did that What does that mean? So what we need Is the rule What is the rule? The rule says If your chi-square is that If it's greater than Your chi-square Crit Then you're going to Reject Your null hypothesis That is the rule That's one thing that we need to Remember that Remember your null hypothesis Which state that They are independent Let's put it in that way Null hypothesis in the Independent And your alternative Will state that they are Dependent Okay Number B The observed frequency of row 1 And column 3 So it means you go to Row 1 column 3 Is the observed frequency 179 you need to Check that Since the p-value Is 0.845 It's greater than 0.05 The two variables Are dependent If we make Decision based on The p-value, the rule says That's another one that you need to Take into consideration The rule says If your p-value Is less than Your alpha You reject The null hypothesis That is the rule That you need to take into consideration What will be the decision Are you going to reject The null hypothesis or not reject The null hypothesis Number D The degrees of freedom The number of rows minus 1 Times the number of column minus 1 But they also gave you the Antithesis Is it that? I don't know Number E The observed frequency of row 1 And column 1 Is equals to Is it equals to 174 I don't know Which one, what are we looking for Is incorrect We're looking for the incorrect statement Do you know which one Is incorrect Come on This thing is not making anything I think option B This Option B Option B It might be that this question Was asking for the correct answer But this option B Says row 1 Column 3 Is 174 It says it's 179 So option B Be the incorrect It might be a surprise Of our lives Is it correct Because I think There is an error on this question Thank you There is an error on this question This should say The expected value If I calculate I think that is why It's correct The incorrect answer will be between A and B A and C actually So there is an error here With the typing of this Expected frequency Not observed So let's calculate the expected frequency Of that Option We will see that it will be 179 Check with me Don't do this to me So the expected frequency Of row 1 column 3 Will be 2000 times It will be 2000 Times 358 Divide By the total of 4000 So when we calculate that What do we get 2000 Times 358 Equals Divide by 4000 I think there is A typing error on my question I will fix that It should say Expected value Now the Incorrect one is between Those two options The incorrect one Will be between Let's try again It will be between How we answer A or how we answer C So let's go back to A and C And Can I bring back my rules Okay, so let's look at the rules So the first rule Will apply to number 1 It says Since our Is the value Our test statistic Is 0,3 And it's less than 0,5,99 So If it's less than We do not reject the null hypothesis So that it means number A is correct It says since The test statistic Is less than your critical value The null hypothesis Is not rejected That is true because Based on the rule, what does the rule say It says if it's bigger We were going to reject So it's not bigger So we're not rejecting the null hypothesis So A is also correct Let's look at C C says The p-value It's greater than 0,05 And therefore it means The two variables are dependent Then it means We are rejecting the null hypothesis If they are dependent Because we will say that The alternative is true But that is not the case If the p-value is less than Alpha, we're going to reject The null hypothesis and state that The two variables Are dependent Or they are not Independent Right? They are not independent Or they are independent So C is the Incorrect one I must apologize One question that we couldn't even See the answer And it still says incorrect That is the one that Value that is incorrect My bad Then there is something wrong that I did with this question But this is the Incorrect option Let me see what the incorrect option Would have been On this question That we need to fix It might be that we don't even have An incorrect question Okay Incorrect Incorrect Incorrect Observed This is incorrect My bad Incorrect So that when you do your activity Then it will be Corrected One This is the Incorrect one I apologize for that one So that is the incorrect one It's safe We can preview and see Since my pictures are not showing It's still fine And I can check And so There we go Alright And that is Moving on to the next one I hope the picture Oh gosh the picture is also not showing So this is the regression Relation And this is the question This is not on We don't believe this That is what is happening On these questions We don't know Why it's not picking up Because the picture is there Okay There is no other way that I can solve That problem It's not allowing me To solve The problem Let's use the picture To answer the question So You can also use The template to answer the question So let's go ahead and use The template to save our time So the template Is the regression template So we just Going to change the values There are only six of them When you use this template You need to pay attention to The instructions So you will need to highlight So there are one, two, three Four, five Six, so I need to delete All these other rows So you highlight from B The rows that you are not Going to be needing up until You get to the X squared At that point You right click And you delete and you say Up And they are all gone The only values you need to Substitute are those two rows I'm just going to Remove the rest And complete the rest Of the rows And then So that will be four Two I'll just enter the X values Eight And you need to pay attention to the Header It's very important Eight Eleven Twelve and sixteen Going on to the next one Is 22 20 19 17 16 13 If I had the time in the world I would have captured this Manually on the question papers Because of my laziness This happened So Once you have all this information We need to answer all the questions Because it's based on the regression You just move to the right And the blue area That's where you will answer the questions So we can start With We can start with the white intercept Remember If you don't know which one is the white Intercept So let's also move this bigger On the right side They are All the information there is your Intercept So you know that the intercept Is this line 40 And you can Minimize this So that you are able to see One of the values Right So there is your intercept And we're looking for the incorrect Statement So it says the intercept Is 22 0.874 And we can see That That And the slope And you go to where the slope is And this is for decimals So you can also Adjust your decimals So they are the decimals So if I adjusted my decimals You will see that they Align With The values I have in front of me You need it I can reduce them There we go This as well Increase the decimals By one point Then increase And a little bit And when you do that Because some of these things are just pictures They just Go on top of the others So There is your Number A is correct Based on the information we have A is correct Then we move to B B says A is strong Negative linear relationship You look for the R's And it will tell you Whether is it a negative relationship Or a positive relationship And then C Says the slope for the Problem above implies that For every Unit Increase The score will decrease Increase means Positive Decrease means negative So the sign in front of the slope For every additional unit Is it going to increase Or is it going to decrease That's how you will answer C D You need to verify Whether this is the regression line And you can use The regression line equation here To validate if this Is the same as what you have Pay attention Remember the slope Multiply with the X The intercept is a constant If stays on its own If you look at this equation Does it reflect that But you can also look at this equation And see if it's the same See The sample correlation coefficient Is equals to Negative 0,93 That's Is it equals to negative 0,93 That is R So let's go to our Question What do you It's the same Questions even though they are Moved around So I'll bring in Bring in that excel sheet So that you can tell me which one is correct A B C D E Which one is incorrect The regression line Is the one that is incorrect Because it should say 22,87 Minus 0,57x And you can see that They swapped around The intercept end And the regression line together And the last question Of the day So it's not wiki I'll just bring the The equation The question So the question says An estimation Of the relationship between Annual bonus Y In thousands of rents And the years of experience Produced The following regression equation And they gave you The regression equation So here you don't need the template There is no way that you can answer This question with a template They also give you The coefficient of determination Of 0,49 Which is your R squared And they say The objective Is to estimate the annual bonus Of a person With 5 years of experience So it means where you see X You're going to put 5 And calculate what is your Estimated bonus And then you can also answer All the questions that they are asking So with option number 1 All what they want you to do Is because they gave you The coefficient of determination R squared Which is 0,49 How do we find R It's just the square root Of 0,49 That's to answer Question number 1 Question number 2 And also bear in mind That Your R In terms of R squared You will never see the side To look at the slope Of the equation To recognize whether the sign Should be negative or positive Because the answer here Will always be positive From R squared to R Will be positive But you need to look at the slope sign To tell you whether Is it going to be a negative R or a positive R For number B Interpret the slope For every one additional Unit of Experience The annual bonus Is it going to increase or decrease Remember plus means Increase On the slope And minus means Decrease So if we add one more Does it increase the annual bonus Or does it decrease That's number C Sorry B Number C There is a positive correlation Coefficient between the annual bonus That you will determine Also The same With the slope Is it positive or negative That is another one Number D The minimum expected bonus Will be 933 Regardless of the number of years So what they are saying is Remember your annual bonus is in thousands It's a thousand of rents So if This is zero Regardless of whether the number of years So if this is zero What will be the average Bonus that a person will get Every year You will meet and then The answer there You will multiply by a thousand And that should give you what you are looking for The Number E They predicted Annual bonus of a person For five years Experience will be 11,514 So you need to take The value of five Substitute it into this and solve this So which one is incorrect I will just give you a couple of minutes To do some of this calculation Remember G Is where X is equals to zero That is the value of your Y hat Multiply it By a thousand It should answer that one For E So where X is equals to Five That is the value of your Y hat Multiply it by a thousand So it means from the equation That they gave you, you just multiply that So I have given you All the information you need So we can go back To I can't go back To my question If I don't Write the equation here You write the equation Y hat Is equals to Zero point nine Three Three Three Two point One one Three X So that you are able to choose the correct answer They use our Are we winning Do we know which one Remember for A We need to tell whether is that an increase Or a decrease Is that correct or incorrect And we're looking for The in What are we looking for We're looking for the incorrect Question or answer Is it B Considering the correlation coefficient has to be positive Is the slope positive Is it B Yes Because it says it's negative zero Seven zero And if you take the square root of The square root of Zero point four nine It gives you Seventy percent But because the slope is positive Therefore it means this value as well Is positive So B is The incorrect one But do you understand What is required of you to answer Some of these questions in the exam You just need to Apply your mind And at some point You will also need to make some calculations As well One more Session For today because 25 is Is just the general Question that I just asked Because I was lazy Now to find more questions as well My Unisa Maybe probably around This time of the day Unisa run some updates Else their system is so slow So the last question It's actually just A general question just to see So Check if you guys Understand if you guys Are happy Because it has nothing to do With statistics And that concludes Our engagement Otherwise There is no Other thing that I need to Discuss any further With anyone Unless if you have any question I just want to see what the question Was all about But it's taking forever But It just says Thank you for being a dedicated student And participating in all the activities That helps you prepare It clearly shows that you are Taking your studies seriously I wish you all the best with your exam Remember You can retake these activities As many times as possible Until you feel that you understand Most of the content As your number one cheerleader And supporter I'm very proud of you And I know you can do it And my last question Of the year Until I see you again next year If you are still doing that True or false Did I meet your expectation For the support I provided For this module Are you happy Are you satisfied And that was the last question Of the year that I have for you And There is nothing more Nothing less that I can add I think even my voice Is very exhausted From me talking since from April much April Until now I need some rest Thank you guys And Thank you for being spot For being here And for always participating Nothing more I'm done Thank you very much for your time No problem Thank you so much Lizzy Thank you so much You are welcome You are all welcome Thank you so much Thank you so much Lizzy Thank you Bye everyone Bye