Good morning, in the first session we looked at how similar attitudes determine attraction and we learned a number of things. For example, similar attitudes lead to attraction, then the issue became whether similar attitudes lead to attraction or dissimilar attitudes lead to repulsion. Then we came up with the idea of similarity dissimilarity asymmetry. Then we came to the question what produces it, is it person positive bias or is it waiting of similar and dissimilar attitudes. And we said both may be operating and when the waiting is taking place it is operating at the level of attention, this much we accomplished in the first session. So, in the second session the only difference you see is between how and why. So, today we are going to discuss why do similar and dissimilar attitudes determine attraction. So, this is our central theme and for this purpose I have organized this lecture in six different parts. In part one we will talk about how the existing model and method and analysis was guided by this stimulus response view which was prevalent in psychology at that time. In the second part we will look at how the model method and analysis guided by this stimulus organism and response that is SOR view guided. Now, in the third part we will come and discuss that like in second part each one thought one thing is operating. In the second part third part when we come we will look at what would happen when all three are operating, but model method and analysis remain of the SOR view. Fourth part I am raising a new issue which would be whether these implicit variables operate simultaneously or they build upon each other. So, that is the issue of simultaneous versus sequential mediation. In the fifth part I will like to put different things in a order and the main message I would give that if there are several mediators like trust, positive affect, inferred attraction, respect. So, if the four things are operating what is the order? So, the message I would be giving would be trust, succeeds in further attraction, but precedes respect and positive affect. And in the final part we will talk about what I have learnt out of this entire research program and we will have some general comments like this is the way I have planned this second session. So, let us begin with the main task here now. So, in part one as I said we are dealing with how SOR model give us some kind of method and analysis to study attraction. So, originally when Bern started his research he came up with this reinforcement model of attraction which was essentially a classical conditioning. So, in his view similar attitudes were like reinforcements. So, which is unconditioned stimulus or UCS and he said that positive like latent variable or unconditioned response is affect. So, if somebody agrees or disagrees with you you feel good and bad.  So, his idea was that pairing of person with similar and dissimilar attitude which you see here at the bottom CS. So, a stimulus the person becomes the CS similar attitudes are the UCS positive feelings are which is originally the UCR becomes CR and attraction is a simple association that they formulated a simple learning framework classical conditioning and they articulated it. Now, how why this view was accepted at that time because we have a reason was at the top and what did we have reason say it was influenced by Harvard professor B. F. S. Kiener who said if psychology is ever to become science we have to study effect of stimulus or independent variable on response or dependent variable without postulating anything inside within the organism. So, for him organism was an empty variable or mind was a black box he said to be science you do not have to deal with unknown you have to deal with known a stimulus we know response we know relate the two why should we infer anything. So, for him human mind was a black box and he started analyzing like this and it had tremendous impact stimulated by Pevlev on the one hand and John B. Watson on the other hand who as father of we have reason. So, because of this SR point of view psychologist had a unique or a specific method of analyzing things which you can see here now in brief I am putting. So, what is our approach our independent variable becomes the X and our Y observable variable response is Y. So, we will manipulate X and examine its effect on Y which is this C. So, if we manipulate X and we measure Y what is the effect of X on Y we will capture by the variable C. So, the C which is the total effect of X on Y whether it is a statistically significant in analysis of variance T test or any test of significance as long as we have significant effect whether it is main effect interaction effect depending upon how many IVs we have that became our goal demonstrating it. So, to test even latent variables for example, in reinforcement model of a variable attraction effect is a latent variable. So, if effect leads to attraction to a study we have to manipulate effect and consequently the pre-paradigm like at earlier stage up to 71 this was the method see here to test this reinforcement model we have to perform six different kinds of experiments. Let us and I call it six different causal chains let us look at the first chain. In the first chain we manipulate attitude similarity which is IV and we measure attraction dv which we discussed in the previous session. The second thing is what is the definition of reinforcement anything which changes the probability of response in a learning situation. So, we are manipulating reinforcements and seeing whether it would change learning. So, if attitude similarity reinforcement what would happen if we use similar and dissimilar attitude as reinforcement it should be able to effect learning. So, in science article was published by Dan Bern lightly and Bern in which they demonstrated that a similar dissimilar attitudes do modify the probability of response in a discrimination learning. So, they did learning experiments at the fourth causal chain we have now attitude similarity lead to effect because this is what we are post-itulating u c s and u c r notion. So, if you manipulate attitude similarity would people feel happy and sad that became the fourth causal chain fifth one became when people get reward and punishments do their mood change demonstrate that. If people feel hot and cold would their mood change that kind of experiments we did and finally, if you manipulate directly effect would that lead to attraction and repulsion. So, to test the reinforcement model we were required to perform experiments which should fall within these six different causal chains it was purely a stimulus response point of view. Now, let me go back to some test. So, in my one paper I did manipulate similar and dissimilar attitudes and measured mood immediate mood of the participants and our measure of mood was very simple at the time we did not have a measure of effect. So, we took item from semantic differential of us good and items were sad happy high low negative positive bad good unpleasant pleasant comfortable uncomfortable on a seven point a scale. Our idea was people should feel happier when they are exposed to a partner expressing similar views than the similar view very simple simple t test type condition. And as you can see that we got the difference between dissimilar and similar here that the mean is higher in similar conditions than in dissimilar conditions on that basis we concluded that similar people did feel happier when the partner shared views than when they disagreed with the participant. So, attitude similarity does lead to effect, but this evidence is again suggestive it is suggestive evidence. So, let us come back to part two now in part two we are coming by with an alternative point of view how psychologist would be conducting their research and that view was also initiated at Harvard university. And the leader is Robert Sessions Woodworth who existed before B. F. S. Keener who had given this idea, but somehow we have a reason had suppressed this idea and we had become more a stimulus response oriented one. So, his ideas can what he said that we cannot ignore organism between a stimulus and response we have to bring it back. And his position was that the same stimulus produces different response in different people. So, the organism is processing it. So, if you strictly take this point of view this implies that organism is a moderating variable same stimulus has different effect in different group of people. That means organism is a moderating variable I am suggesting something else that if organism we treat is that over really represents the latent variables not necessarily moderating it is it represents the effect of a stimulus internal processes in that case organism can be consider as mediating variable. And these points of view can be found in these references in which he said like a beautiful thing you would see like his first book was psychology a study of mental life a skinner called it black box. And the two other books which actually influenced me in my thinking in becoming a psychologist I have not come across books as influential as these two editions of Robert session Woodworth. So, those who are interested in psychology if the library still has this copy one should at least glance through it why these two books are considered so powerful. Now, after Woodworth there was a revolution in psychology and three pioneers played a major role Donald Broadbent of Oxford, Norm Chomsky of MIT in language. And Ulrich Nasser from Europe these three people and I have given their books which produce tremendous impact in the revolution of what I am going to discuss. They basically brought the information processing approach to human behavior. And they said that dynamic information processing system we have to study and mental operations can be represented in computational terms. They also said that what happens between a stimulus and response we have to analyze in that how this sensory input is transformed reduced elaborated stored recovered and retrieved or used. We must study it in other words their position said that we must specify the variables that intervene between a stimulus and response. And that lead to what I am going to describe now. So, in their view considering the latent variables are not only necessary, but also important if we want to understand psychological processes or psychological effects of any manipulated variables. And that brought to a new revolution in psychology. So, here I present you a general mediation model for which the pioneers bear to social psychologists from university of Connecticut, Baran and Kini. Let us look at the model here. In a stimulus response view we have basically independent variable and we have dependent variables. What we were looking at that a small c does this i v have any effect on d v. Cognitive psychologists the three point I suggested or Robert session would worth idea they said that that is not enough. We also need to study what carries the effect of i v to the d v and that they termed mediating variable. So, to do that we need to measure one more variable and that is the mediating variable. So, they said that we need to have a path from independent variable to the mediating variable that we call path a. So, originally we will have analysis of variance in which we will test effect of i v on the d v or the six chain you will go effect of i v on the m v. They said no we can put them together we need not analyze them separately. And what is the beauty you see here this c we are now partitioning into two things. One do two regression analysis in the first regression analysis we are predicting d v from this i v. And in the second regression analysis we are predicting this d v from the i v and the m v. So, from the second regression analysis you are getting this v which is effect of m v on d v when i v is present and this c prime is effect of i v when m v is present. So, he said estimate two more things then do another simple regression in which we are examining effect of i v to the m v that would give us path a. So, now we have three new things analysis of variance would give us only c when we do regression analysis we are getting c we are getting c prime we are getting b by doing another simple regression we are doing path a. So, when you look at these four a b c and c prime the picture is clear.  Which analysis of variance would indicate in form of a significant difference c is the direct effect of i v on d v when effect of m v is controlled. And if you take product of a and b this is the indirect effect of i v through the m v. So, in the in the bottom you can see we can have three different checks on this c would be equal to a times b plus c prime or c minus c prime would give us a times b or c minus a b would give us c prime we can check with one another it would be the same. So, this technique actually allowed us to partition c into two parts what is the effect of i v and what is the effect of i v through the mediator that is all is mediation analysis. So, this one you say a times b is in the circle here and the reference for this is this baron and keny 86 article in which they gave idea for the mediation analysis. Now this is what I am going to suggest now. So, first mediating test was by kandan and krayano in 88. So, though baron and keny had proposed in 86 still it had not produced impact. So, how they look that it they manipulated attitude similarity and measured ask the participant can you tell me how attracted the partner would be toward you that we called inferred attraction. Then we said can you tell how attracted you are toward the partner that is the attraction part. So, two measures we have and one i v we have that is all and they did some regression analysis. So, if we correlate i v with d v the correlation is 0.64 and significant, but if you correlate i v and d v and control the effect that is like the second stage regression analysis that correlation becomes 0.18. So, the direct effect of attitude similarity on attraction is 0.18 total effect comes to 0.64. So, this reduction he said is the mediation part by inferred attraction. So, since this effect is totally significant and controlled effect c prime is significant he said inferred attraction is a partial mediator it is not the complete story it is a mediator, but incomplete it has effect, but it cannot explain fully. Another thing he did they did and what they did he said inferred attraction and attraction correlation is 0.81 it is m v and the d v, but if I control the effect of attitude similarity what would happen that correlation could be reduced only to 0.66. So, they concluded that look inferred attraction is a greater factor in similarity attraction than attitude similarity by itself. Here is one issue to notice which I followed up little later. In 2004 another attack came and I would like to draw your attention as I mentioned in the first lecture that sometime measuring some irrelevant things can be problematic to you. Let us look at what he they demonstrated they said that no we feel attracted towards the partner because we have respect for the competence of the partner. We think he is very competent person and I respect him that leads to attraction. Moreover burn initial items of intelligence and general knowledge is a measure of intelligence. So, when you measure it you are making cognitive evaluation salient. So, if I measure attraction without it and with it there should be a difference effect would be a stronger when you measure after cognitive evaluation than before cognitive evaluation. So, you see like what was considered buffer items became a basis of new theorizing. So, they did it and made that prediction and here are their findings. First how did they measure his measure of respect was my partner would make a good leader will achieve all of his or her goals is good at everything he or she does will probably be successful in life essentially competence part. Attraction he measured not my working together and liking here I would like to meet my partner I look forward to meeting my partner I look forward to working with my partner I would like to get to know this partner better. So, it is more like a behavioral attraction you know like in tension to establish relationship acquaintance you know that kind of notion. And here is the effect of order of presentation two lines I am showing you three conditions we have dissimilar attitudes no attitude information the condition we created had created and similar attitudes see the line here the line with open circle and solid is when cognitive evaluation was measured before attraction. And the dashed line you see is when attraction was measured first in this experiment it was really not significant later on we have found there is effect, but the effect is not as strong as when attraction is measured last which is natural because if attraction is the dv last measured variable mediator should have some effect on it. And this is what you are not seeing another thing Mantua and Horton did when attraction was measured second they did mediation analysis. And that mediation analysis is just like I had shown Baron and Kenny you see here attitude similarity has effect on attraction 0.54 attitude similarity has a respect effect on respect 0.59 C prime is 0.18 which is non significant now. And respect has effect on attraction that is the b path. So, if you take the product of the two that is coming 0.236. So, 0.54 minus 0.36 is giving us 0.18. So, on based on these he claim that respect is a complete mediator of attitude similarity attraction effect one more thing I would like to draw your attention. At that time simply saying that when you enter m v it reduces the effect of I v is not enough reduction must be greater than 0. So, for that purpose you see we have a sobel z test this should be significantly greater than 0 and he found that it was that z test was significant. So, on that basis he said respect is a mediator and it is a complete mediator. Now, but one thing they wrote in their discussion part which I would like to draw your attention. They said that purely cognitive mechanism effective mechanism is not needed. Nevertheless, I quote them they admitted that self protective self protective motives guided decrease attraction to an exceptional individual who could evaluate the self negatively. So, if you feel there is a competent person who can pose threat to you we do not feel attracted toward it.  So, he did they did not rule out cognitive mechanism. So, I tested effect as a mediator. So, manipulate attitude similarity major effect and major attraction and here are my findings attitude similarity leads to attraction 0.55 attitude similarity leads to positive effect 0.38 when we put them together then effect has 0.40 and direct effect comes down to 0.40. So, if you take the difference and the difference of 0.15 is again significant. So, effect also reduce the effect of attitude similarity as if it were a significant mediator. So, now, we have evidence for three mediators inferred attraction respect and positive effect and this we had we published in 2007 here where we said effect is a partial mediator. Now, based on these literatures here are few points we can make one all the three formulation seem to be right. That means inferred attraction respect and positive effect are potential why they can explain, but there is something intriguing too. How come mediation was partial in first and third case, but it was complete in the second case. Another issue if you do one mediation test and if you get that the mediation is complete does it mean that we should not be considering another alternative issues. These were the challenges and that brought me to this idea and that idea in Oxford free India reader book four in my high school I had read it that six blind man of Hindustan wanted to understand elephant and they went and touched different parts of the elephants and they thought that elephant looks differently to somebody it was a snake to somebody it was a spare to somebody it was fan to another it was tree to the fifth one it was wall and to somebody it was rope. I thought the upper predecessors including myself we were no different from these blind man touching different part of the same phenomenon same attitude similarity and saying this is the truth. Then I went back to internet and I started looking at ideas of some of the physicists and one idea which caught my attention in particular I would like to share with you what we observe is not nature in itself, but nature exposed to our method of questioning and the man who said is Werner Eisenberg who got Nobel prize in physics and also he was Max Planck gold medalist. Now, think about it that at no point of time we are knowing the truth or we are knowing the nature our method is allowing us to have some understanding of it this is the message I am giving. So, what should we do it what should be our research strategy we thought of this and that brings us to part three now what would happen if we bring all three mediators together within the same stimulus organism and response formulation let us do that. So, I my students and I started doing multiple mediator tests and in the first study we measured all three positive effect respect in further traction from attitude similarity and attraction was d b and two orders we manipulated. Since respect was always before attraction you can see that we are putting respect before, but we varied the order of positive effect and negative effect that we are manipulating, but this order we are keeping constant when we collected the data we got something like this and what you notice here one more dummy variable I added can you infer how much your partner would respect you we are asking how much your partner would like you is in further traction, but how much your partner would respect you we built additional thing and why I did I will tell you little later. So, when we plot the same analysis we have earlier we will have one mediator now we can have four mediators at stage two regression analysis and let us look at it now and product of a times b for all four I have written this side. So, let us look at this one first when we look at inferred respect you see path a point one five is non significant and also path b point zero nine is non significant. So, if we added dummy variable using the same method it is not playing any causal role. So, methodologically it is important that theoretical variable should mediate not the dummy ones. So, this is methodological point I have scored then let us come to positive effect and here is something very intriguing this effect is marginally significant, but it is negative theory predicts positive, but this one is negative what do we do in a situation like this, but when we come to respect and inferred attraction the effects are positive and significant. So, this study though it was published it opened a pandora box and the point which I would like to hammer here in a spite of the three mediators the mediation is partial c prime is not non significant. Mantua and Horton one was enough here in spite of three mediation is not complete. So, another thing you which I said effect has a negative it has effect, but it has negative effect if you feel bad then you feel attracted that kind. So, how do we explain here is a challenge. Now, because of this we did another research in which we do did two things we manipulated inferred attraction and positive effect and we manipulated inferred attraction and respect take two at a time in both cases. Now, you see here inferred attraction and positive effect mediation is partial, but when we put inferred attraction and. So, in this case it is a partial mediation, but when we put inferred attraction and respect mediation is complete. So, we have some clarity and interesting part is the effect has a positive effect in earlier it had a negative slope now it has a positive slope here. So, we thought the issue is resolved, but my mind did not become satisfied with it we thought something else is operating and we must visit this. So, from multiple mediation what did we learn number one yes my colleagues or my we were right in saying that this one is a mediator there is no doubt about it, but we were also wrong in saying just like the blind man that this is only mediator effect is mediator, respect is mediator, inferred attraction is a mediator we were mistaken like the blind man that this is the answer a wall a rope a tea or a snake that was our so why partial mediation even in a multiple mediator test is another intriguing issue. So, we thought the phenomenon is still misunderstood our understanding is not complete that led me to do something which are not published, but hopefully they should be and I share with it. So, one issue becomes issue of simultaneous versus sequential mediation that when we have more than one mediators should we examine the relationship among them. Let us come back to come to the fourth part here we looked at the literature and we thought that models can be we can develop several models one model is the first one which I had tested in 2007 that inferred attraction the respect and positive effect which I say model one simultaneous model this simultaneously carry the effect of attitude similarity to attraction. Byron Bern and Bransbeek in 2006 edition of social psychology they thought that no the three can be accommodated something like this and what they are saying is they called it effect center model of attraction which is simultaneous and sequential and they said attitude similarity lead to all three, but inferred attraction and respect come before positive effect and they also boost positive effect which leads to attraction that became effect center simultaneous sequential model. We thought that no this is not complete the correct model can be third in third one we say attitude similarity it has direct effect you see solid line to all three plus there is a sequential effect inferred attraction leads to respect which leads to positive effect which leads to attraction or it could be positive effect respect or inferred attraction. So, if you go back to the literature we have idea for these these two are sequential model so third model is the effect centered one I am retaining effect as the last position but instead of considering them as simultaneous I am saying they are also sequentially linked when we come to this one Jayank in 1980 American psychologist he published a paper feelings need no inferences his idea is effect is the first process in any reaction you know without any control spontaneous. So, based on that we generated model three and four so you see we have a theory simultaneous mediation then we have effect centered we revised effect center and we also thought Jayank model and to do to test it we conducted two experiments now the first look at the challenge here when we are doing any study with three mediator the first challenge is to demonstrate that each one is a mediator then you have to put them together and see how to the mediated this is the one issue another issue is when you are measuring them how do you know that they are different things are not the same thing. So, each your m v and d v distinct construct if they are the same thing then it makes no sense doing it. So, the first task was to demonstrate whether these things can mediate when we have because same method same participant population we have to do. So, the first study we did and you can see here that positive effect did mediate and that now by this time there is another change in the method earlier your a times b should be greater than 0 was being tested by sobel test people realize that our indirect effect has no normal distribution. So, sobel test is not the right test. So, now another technique has developed this is called bootest sampling what we tell the computer from the same data 1000, 2000, 3000, 5000, 10,000 whatever you want draw the sample and from each time develop the estimate the effect and tell me what would be the 95 percent confidence interval of it that 95 percent confidence interval should include your indirect effect, but not 0. So, if it is positive lower side should not include 0 if it is negative the upper side should not include 0. So, this is what I am writing as 95 percent confidence interval. So, this is another so we are passing through almost a methodological revolution these days between 2007 and now I am reporting 6 years later something else. So, see here so when we look at positive effect we have indirect effect of 0.052 and this indirect effect is between 0.025 and 9 0. So, that is a significant and mediation is partial because it is not non-significancy prime when we come to the respect the same thing you are finding when we come to the infrared reaction mediation is complete. So, just like the previous literature something is appearing complete something is a partial. So, that give me a ground to say that yes these 3 are mediator and they can be considered as a mediator, but on another thing this is what I had shown by the circle. So, we did find that they are, but an important issue it became since third one is a complete mediator does it mean that we should not consider respect and effect my answer is no even though somebody is very powerful it does not mean if professor misrise very powerful it does not mean that we are not effective we produce some effect this is the logic we have. So, question of construct distinction here now how do we test whether positive effect and attraction are same thing or different thing. So, for this we do confirmatory factor analysis and we make two models we tell the computer treat them as two different variables or treat them as one variable. So, the first time we are saying we are getting a chi s square value second time we are doing when we say one variable we are getting chi s square 37 point something when we say second variable 61 and difference between the two chi s square is significant. So, we are saying that a two factor model is more appropriate than a single factor model this is in case of the first design in the second case same thing you are finding that chi s square difference I am showing by delta chi s square here in the red third time also we find found the same thing. So, the two constructs I V and D V bear seemingly different they were not the same thing for the second experiment where we measured the three things and attraction again you can see when we said four factor model 2.71 chi s square and single factor one we have a nine point something here nonsignificant chi s square really means your model is successful lower chi s square means your model your model is not perfect, but of the two defective model your model is more preferable than the other one and that is why this chi s square difference we calculate. So, we do have evidence that these variables are not the same thing in 2007 how I had handled this question by including a dummy variable, but now you may not included dummy variable and still you can test whether the constructs are same or not. Now, let us come back the main question here let us look at the model from the data which model can be suggested supported by this. So, first one is the multiple mediator model so, for that we do path analysis one need not be frightened by path analysis it is easier friendly and if there is a most program I can teach it in 5 minutes.  So, now see something here chi s square is 184 significant this is n n f i is called non normed fit index theoretically it should be 0.95 and more then we have incremental fit index it should be 0.095 and more then we have r m s c theoretically it should be 0.05 and less and s r m r should be 0.05 and less if your model is really perfect fine if it is not then we compare the chi s square which one is giving a better fit. So, this one clearly you see that we are not getting a good fit and this is the model I had tested in 2007. So, if I did not get a good fit if there is nothing to be surprising with a new set of data then I come to effect center model of burn in which part of the data I had measured variable precisely in this way and this is from the total data 384 participant that is 128 participants. You see this model fit is not satisfactory either even with the complete set of data the model fit is not very good. In fact, it is worse and more surprisingly burn said in further attraction leads to positive effect that path is 0.08 you see. So, his simultaneous sequential model is not right would the model 3 and 4 succeed then 3 and 4 plus I generated to more model and why I did I will tell you at right time. Let us look at model 3 and 4 they give perfect fit and all path coefficient if you are significant in other words effect primacy and effect center model they are indistinguishable both are giving exactly the same fit. So, you can raise a doubt that your technique is such as professor Dalal has been saying that he can prove anything mathematically if it is so then if I reverse the sequence I should be able to get the same thing. So, for that I generated model 4 and 5 and all I have done I have put respect as last variable as I had done in 2007 and I put respect as first variable and when we put it then see the consequences with the same data by simply changing the ordering the fit becomes as bad as in the previous models. So, this is the one one script which we prepared and I must thank Braj Bhushan for commissioning for me for this because it was not written and I started working and this is what I found out a manuscript is ready in addition to the lecture. Now, let us what we have. So, model 3 and 4 are satisfactory but indistinguishable model 4 and 5 are inadequate. So, clearly respect does not deserve to be proximal or distal to attraction it operate somewhere else it operate somewhere else here. Now, that had lead here is another is there in literature there is also a point that if somebody is attracted toward you then you feel attracted toward him. So, inferred attraction is not the cause it is effect of attraction I also wanted to handle this question. So, the last 4 models I reversed and all the change I have done I have put inferred attraction as d v and attraction as the mediator. So, in model 1 you see attraction there in model 4 you see here in model 5 you see first in model 6 you see attraction as last if it is. So, then the model fit should be very good number 1 and all the path coefficient should be significant. So, my mathematician friend say mathematics can do anything I do not believe it see here model 3 and 4 all the fit indices chi s square significant n n n pi unsatisfactory i f i satisfactory r m s e is unsatisfactory s r it is a mixed up it is not giving good fit, but another thing I will draw your attention. When we come to this it is worse, but worse than the model fit is that whenever you put effect of from positive effect to inferred attraction path is nonsignificant from positive effect to inferred attraction is 0.04 just like in the first case from inferred positive effect to attraction this is 0.04 this is 0.04. So, causal transmission is not from positive effect to inferred attraction it is from inferred attraction to positive effect both ways we are getting. So, from these 2 experiments what do we learn none of these 4 sequential model can allow inferred attraction to be the outcome variable it has to be a mediator. So, this is what first 2 I say 2 models can represent better, but they are indistinguishable both are consistent with the literature and inferred attraction is not the outcome variable it is a transmits some process it is not the final process that much we have achieved. So, from these essentially I learnt 4 different things 3 previously known mediators are sequentially linked to each other. If you disturb it then you cannot represent your data number 2 the 2 defensible model seems to be inferred attraction leads to respect leads to positive effect which is model 3 or positive effect leads to respect leads to inferred attraction which is model 4. So, this model 4 would be effect model of gynec the first one is effect centered model and model 3 which is modified version of effect centered model and model 4 which is. So, when we suggested here that we could not distinguish whether model 3 or model 4 is right can it be distinguish that is a challenge that is that brings us to part 5. So, let us come to part 5 now. So, in part 4 we looked at that there is a sequential mediation and we cannot distinguish between 2 models that is effect centered and effect primacy. So, for that we launched another research program here and the main message I would giving you here we have introduced a new mediating variable trust in the benign violence of your partner that this partner would not take advantage of me would not exploit me that kind of notion and then the message I am going to give you that when we put these 4 things trust respect positive effect and inferred attraction then inferred attraction comes before trust. In other words trust follows inferred attraction but precedes respect and positive effect. So, essentially we dense for effect centered model and let us see how we came to this conclusion. So, a challenge is now to demonstrate that trust is a new mediator. Now, why do we have to demonstrate it if we go to psychological literature we find some views evolutionary psychologist claim that we all are equipped with a cheater detector mechanism. So, when you encounter a person you know that gets activated. So, if you feel that this fellow who is not trust worthy we avoid it. So, this is the one suggestion we have another suggestion we have here which Simpson said that without trust voluntary relationship are not likely to develop let alone grow or be maintained that means trust is important. Another suggestion we have like Cottrell Newberg and Lie they ask people what is the most desirable characteristic of a partner across a wide variety of groups. And they are finding I am giving just quoting them they said people tend to assign trust worthiness high values on likert scales of importance to select trust worthiness as the most necessary characteristic. And they locate large portion of limited resources to increase target trust worthiness in 2007 they said it. Then another thing we found in a long literature in sociology again where if sociologist Goulder said in 1959 people like those who like them. And trust mediate this reciprocated attraction in 2008 Mantua and insco demonstrated. So, we did a published a paper in European journal of social psychology in 2009 where we said some traits can be profitable to other people some traits can be profitable to yourself like basically warm should be example of other profitable. And that effect is also mediated by trust all these led us to consider trust as a new potential mediator of similarity attraction. So, we initiated a research program and I am going to share my findings from that research program. We conducted a series of six experiments in which attraction is always the dependent variable. And everywhere we are manipulating in five experiments we manipulated attitude similarity as I V. And then we pitted trust against one another as M V one by one trust versus respect trust versus positive effect trust versus in further attraction. And then all them together you know something like this we pitted in the last experiment we went back to the causal chain hypothesis from which this attraction research we had just started. So, what we do manipulate partner attraction not that you infer partner is actually attracted toward you then measure trust and attraction in another we said manipulate trust worthiness. And then measured inferred attraction and attraction logic is if the causal chain is from inferred attraction to trust to attraction mediation should be complete there. But, if you have trust inferred attraction and then attraction there should be mediation because they are correlated, but it would not be complete that logic we followed. So, three key issues I am dealing with in this research program number one each trust in the partner variable and say new mediator. Number two each trust distinct from attraction and the three previously known mediators. Number three what is the sequential order of trust among these M V. Now, please go back to my model three and model four from the previous paper where we said two possible orders have been identified so far inferred attraction respect and positive effect which is effect centered model or positive effect respected inferred attraction which is effect primation they were indistinguishable. So, can trust lead to a choice between these two indistinguishable models that these were the. So, trust as a new mediator how we did it experiment one very simple experiment manipulate attitude similarity with three level 0.5 and 1 measure trust in an attraction toward the partner 90 participants huge maximum likelihood factor analysis with pro max rotation to see whether attraction and trust are same thing or different things because we have not studied them together. So, we are not doing confirmatory first we are doing exploratory. So, here are the findings the four trust items you have my partner would look out for my interest my partner would act many more lently toward me this partner would make me feel secure the I would find this interaction partner to be dependable. And you see and the attraction items I would like to meet my future interaction partner I would like to be with my future interaction partner I look forward working on the upcoming task with my future interaction I would like to get know him better like this. And you see the now I would like to also mention here these are not factor loadings they are regression weights. So, you are finding that these four items are having higher regression weights on factor one in contrast attraction items are having higher regression weights on factor two. And this program also gives you a chi a square test where your two factor model is giving good fit or not. So, when I said do my data have two factors I am getting chi a square value of these great could have been if this chi a square should have been non significant, but it is not it is significant, but it is 26.71, but when I say one factor it is 95. So, on that basis I am concluding that trust and attraction are distinct factors they are not the same thing. So, measurement model is satisfied now we come to the causal models in causal models you see all three have both variable have a linear trend trust increases as a function of proportion of similarity tools or traction increases as a function of proportion of similarity tools. So, causal effect is also demonstrated now is it mediator or not. So, we do the simple mediation analysis and you can see that the original effect of 0.079 here we have even a stronger effect because this value is becoming negative and product of these two is 1.0795 percent confidence interval. So, trust is a mediator and it seems to be the only mediator for this data it is a soul mediator, but I am not accepting that one. So, we have succeeded in showing trust and attraction are distinct and trust is a mediator of similarity attraction. Now, we go to the next experiment where we are manipulating we are pitting trust against positive effect at this point you also became interested in something more what would be negative effect we have not considered so far. So, we measured both positive and negative effect trust and attraction here our participants are just 68 here positive feelings by active attentive determined and inspired and negative I am calling as fear because in long relationship fear is a factor. So, we have a scared nervous jittery and afraid and from Panas we have taken items which is original 10 items, but in 2007 Thompson wrote one paper in journal of cross cultural psychology and said that no need to use the 10 item these 4 items are enough and these are the 4 items which have highest loading in some 1700 participant across the world that is why I am using those items. And we measured all the along the 7 point scales here. So, what is the result from experiment 2 let us look at first the measurement model when we do confirmatory factor analysis our 4 factor model 2 emotions trust and attraction this model is doing better than saying that this we have just one factor model they are not the same thing they are different things. And when we come to the causal model here you can look at it what is happening attitude similarity effects trust positive effect and the sequential dependency we notice like this and when we put attitude. So, I have reverse the order here put positive effect first or trust first the question is how do we choose between all paths are coefficient or significance here our choice is based on the model fit. And you can see model 1 gives a better fit and satisfactory fit than model 2 that means in this experiment the order is trust lead to positive effects which leads to attraction not positive effect leads to trust which leads to attraction. So, no effect primacy this is the outcome from the because the second model you see everything is very weak fit indices we have. Then we come to experiment 3 now we are putting trust respect and attraction same logic we have same method we have here just 96 participant 2 levels of similar dissimilar attitudes. And respect items are precisely like those which I had shown you earlier attraction items were same again the construct distinction we are seeing with the trust respect and attraction are same thing or different thing if it n c is that they are different things they cannot be treated the same thing. So, this we call measurement model is satisfied now we are interested in the causal model here is the model. So, we put trust respect and attraction or we put a respect trust and attraction which model would give a better fit. And 3 kinds of line I am drawing direct solid line means effect of I V on the M V or D V then we have a dash line which is indicating effect of a mediator on the D V. And then we have a very thin line which is the sequential effect. So, when we fit this model again you can see model 1 in which trust comes before respect is satisfactory and acceptable model 2 is not. So, between effect and trust trust comes first between trust and effect trust and respect trust comes first. So, that point is also a score when we come to experiment 4 then we are bringing inferred attraction 1 by 1 not all that many because if we put many then we do not know which is the source of trouble. So, always put 1. So, when we put like exactly like same thing we did just 60 participants we have. And inferred attraction we said my partner would like to meet me my partner would like to be with me my partner is looking forward to working with me and like to get me know better means not only me he is also interested in this. When we tested it again you can see that a 3 factor model is a better representation of the data than the 1 factor model. But the causal model is here in which I have switched the order both are giving making exactly the same fit. So, when we put inferred attraction and trust we cannot distinguish which one comes first and which one is the second one this is the challenge. Same problem we have faced in the previous study that when you measure inferred attraction respect and positive effect you cannot distinguish. So, we are back to the same problem when we put this that lead me to go to what professor Mishra ask a study a day before a study about long term relationship. Now, they have a new conceptualization they say in any relationship formation the first goal is to see whether that fellow would accept me or not whether that fellow would accept me or not. So, we thought that inferred attraction is accepting that this fellow accepts me if he accepts me then I signal trust to him and then trust leads to the other things this is the logic we followed. So, for that reason we conducted the experiment 5 in which we are putting all these responses positive effect negative effect trust respect and inferred attraction manipulate attitude similarity measure attraction as simple as this. And 224 participants we have more here because for doing confirmatory analysis is more variable you need larger sample size. And 4 order of measurement of these things we manipulated in construct distinction you can see that a 6 factor model did better than a 5 factor model when we said that these responses mean 5 things it did not give a good fit when we said 6 it is a better fit. So, that job is done now causal model we are testing here. So, model 5 and 6 here only difference you can see I am putting trust inferred attraction respect and positive effect which is essentially effect centered model with trust coming before infer. But when we put inferred attraction trust respect positive effect order only in the first two I am changing I was as expected model 2 is giving the bit not perfect fit, but it is giving a major fit idea is inferred attraction is the clue that that fellow accepts me. So, trust is a signal which leads to respect which leads to good feelings and finally attraction toward the person this is what is happening in this case. So, that is why I said trust succeeds inferred attraction, but precedes respect and positive effect that much we seem to have achieved, but still any experimentalist would not accept the cause trust inferred attraction respect effect all are correlated variables. So, that is why we design experiment 6 in which we said flow should be a smoother and a stronger from a cause to the effect than from an effect to the cause a simple logic we are following from every designs which I learn in my high school. So, when we do this then partner attraction should lead to trust which lead to attraction this is what 1 2 and in school in 2007 and demonstrated I am simply borrowing to my, but I am reversing it trust then inferred attraction and attraction here flow should not be a smoother it should be weaker this is the this is the point I want to demonstrate and. So, d v i b d v link should be mediated fully at chain 1, but not at chain 2 because it is not the natural flow and you can see here that this is again we are back to the single mediator test and you see here in this case effect of partner attraction on attraction is mediated fully by trust the top graph, but when we come to the second graph effect of partner trust worthiness on attraction through inferred attraction which has already taken place before that one is partial. So, we have a scored a point here the trust seems to be a successor of inferred attraction, but predecessor of respect and positive effect here this is so as we had predicted chain 1 had a full mediation at chain 2 we did not have a full mediation. So, what we have learned out of these 6 experiments attraction is distinct from trust inferred attraction respect positive effect and negative effect I think after inferred put the attraction word it is missing here. Then when trust was measured with either positive effect respect it was a predecessor when we measure them with other potentials mediators then sequential order was like inferred attraction and trust became ambiguous you cannot tell them, but when we put them together like a right pecking order you have to have the complete population. So, when we have it then trust does succeed inferred attraction, but proceed respect and positive effect which supports and when we manipulated experimentally the causal flow is from inferred attraction to trust to attraction not vice versa this is what we have demonstrated. And surprisingly effect negative effect was neither influenced nor a influencer or mediator of attraction it was a simple correlate in sometimes this is which is another challenge we are finding. Now, overview I am giving you now what I have learned and what would be some general comments few things which I feel that we have succeeded one we have succeeded in identifying the sequence of the mediator. So, we are offering in the our contribution to that paradigm is we offer trust as a new mediator and we specify the sequence among the already known mediators of the similarity attraction link this is the contribution we have made and that sequence seems to be inferred attraction trust respect and positive effect clearly then effect is proximal to attraction as original 90 70 model of burn and clore had said we did not have a method to test it, but when we have a method we have evidence for it. Second thing what is meaning of inferred attraction and trust again from close relationship plus this data I find that in inferred attraction is an impulsive I mean automatic natural spontaneous appraisal of acceptance whether you accept me or you do not accept and trust is a signal of continuing the interaction. But, Bhushan has accepted me I want to enter into relationship Kiristhor has accepted me I want to enter into this is the interpretation I am giving, but trust seems can be impulsive or reflective like in experiment 2 and 3 it was like a natural response when there is no inferred attraction, but when you put inferred attraction then you think about it he has accepted me now I can say that yes I am interested in the relationship. So, in that case it is becoming and you know so in that case I am saying it can be reflective that means automatic evaluation part versus something which is activated by cheater detector would be the impulsive part, but this one is a considered cold light of reasoning he is accepting me and I should be entering into this relationship and there is a safety in approaching so and so in that case it is operating. So, trust of the soul is more reflective by Salvi is saying similarity attractionally is more reflective than impulsive at least in my experiments we can link with impression formation. So, how do we link? Now, Sujan Fisk of Princeton after reviewing the literature on impression formation they concluded that people have a tendency first to determine the intention of the other person and group and second their ability to act on those intentions like Fisk, Kadhi and Gillik advances in experimental social psychology they have a chapter. Now, is it trust that intention part and the respect for your ability is whether you can do it or not I am linking that and this is what we said the trust comes before respect. So, I am linking that attraction research. So, our finding the trust precedes respect in similarity is actually an important in the case of external important convergent validation of that kind of research. So, that way I am linking the two paradigm what is happening in social cognition and what is happening in interpersonal attraction situations they seem to be the same the same sequence of trust and respect can represent it. So, personality trades or attitude they may be producing similar kind of effect this is another thing I am mentioning few general comments I would like to make in 1992 Doc and Barnes reviewed literature on attraction paradigm and they made this observation the debate about similarity and attraction has been the inverted intellectual titanic of last 30 years everyone thinks it should sink, but it does not. Now, in 2013 this is 92. So, 821 years later I am telling you that what I have shown that this link is a steel buoyant for over half of a century go by to 61 the first publication and two in 52 years a steel we are not able to understand we have come to this point and I further predict that interest in this link will remain as long as attitude per se remains the central and most indispensable concept in social psychology Wigner and Carlston in 2005 there is a book in which they have said. So, I am saying that as long as we have interest in attitude this relationship cannot die we will be dealing with it and when I observe the prime time I get more impressed that contingent upon your attitudes how people argue, support, reject, condemn and I have added few important references not all, but here are few references from which I based this work. So, two slides we have these references and finally, I thank you for your attention thank you.


Welcome to our review for exam one for math 1060 trigonometry for students at Southern Utah University. As usual I'm your professor today Dr. Andrew Missildine. What I want to do is very briefly in this video talk about the types of questions you're gonna see on exam one. Now there's a lot that goes into preparing for an exam. One of the most important things is to know the time, place, and manner of set exam so that this video can be reused semester after semester. It will not include information about where, when, the exam will take place and some of those other more semesters since to semester specific policies. You are encouraged to look at the course syllabus, the information on Canvas or just talk to me, your instructor, to find out those specific ideas about time, place, and manner of the exam. Now there are some important things that will be the same for all exams throughout this whole semesters and I do want to establish some of those things right now. So with our exams this is a paper exam. You should write on the exam. There will be two types of questions that you do see on the exam and they'll come into two sections. There's the multiple choice section and there's the free response section. The multiple choice, the multiple choice section as you might assume, these will be multiple choice questions that you'll be given a question and you know four, five, six possible responses. You would then clearly indicate the one single answer that is correct. Without marking an answer or if your answer is unclear, you would get no credit for that. But if you select the correct response and no others, you would get full credit for that type of question. On a multiple choice question, on a multiple choice question, you don't get any partial credit. You get points for getting the question right or you don't get points if you don't get it right. So there's no partial credit available on those types of questions on the multiple choice section. This is in contrast, of course, to the free response section. In the free response section, you must show all of your work, not just the final answer. The final answer is worth maybe a little or just writing the right answer with no work could get you no credit whatsoever. So you have to show all of your work. And that's the main difference between the multiple choice questions and the free response in terms of grading. One other also important thing to mention here is that our exam will actually come in two pieces. In this video, I'm going to show you both pieces together. But when you take the test yourself, there'll be two portions that you can take separately, the multiple choice section and the free response section. This is the reason why on the multiple choice section, you will not be allowed any notes, you will not be allowed any calculator to use. These are questions that you should be able to answer just using basic rudimentary arithmetic. So nothing is going to be too complicated there. These can be on the easier side. It will require, of course, you have certain principles, formulas, memorized in preparation for that multiple choice section. That's one of the reasons why we have this video and the exam syllabus. So it's very clear what it is you need to be studying and memorizing of anything. There will be a formula sheet provided to you for the multiple choice section. This is mostly to provide you with trigonometric identities, which while that's not going to be super critical on this exam, that might be helpful on future exams. Of course, do not write on your formula sheet because these could be reused for other students as well. In the free response section, I should mention that things will be a little bit looser. So first of all, the thing you're probably going to like the most is you are allowed a calculator in the free response section, but it has to be a non graphing calculator, aka scientific calculator. You can't use one with a graphing utility or symbolic algebraic manipulation or anything like that. It's just a standard scientific calculator. You would want one definitely with the trigonometric function, sine, cosine, tangent, their inverses. It would be nice if you convert back and forth between degrees and radians. You should be able to also just use no basic arithmetic additions, attraction, multiplication, division, exponents, radicals, those type of things. So just a basic scientific calculator is what you'd be allowed on that. In terms of notes, you will be allowed a single three by five no card that no card should be three inches by five inches. So 15 square inches of area. You can write front and back of that no card.  And this no card I should mention is actually part of the exam. Technically, there are 16 questions in the 16th question is your no card question. That is, you'll you'll get either two points or no points by turning in a valid no card or not turning that in. So you can put whatever notes you want on that no card. Those can be trigonometric identities, formulas, examples, whatever you think is appropriate. Draw Perry the platypus if you want to. I've seen people do that before. Whatever you need to help you do well in the exam, put that on your no card. It'll be stapled and termed in with your exam when it's submitted. So just be aware. Be aware of those things that is two points out of 100. You're going to want to want. You are going to want to do that no card there. And again, the no cards only available for the free response section, which we'll talk about both. And I'll make it very clear when we go from one section to the other. The other things I should mention. I guess the only other thing I really want to mention for this exam before we get into the more nitty gritties of questions is that there will be occasionally geometric diagrams that are drawn. Do not assume that these diagrams are drawn to scale. That's not necessary. That is, you shouldn't necessarily assume they're drawn to scale. So only use the information that's provided from what the diagram says and don't infer anything beyond that. And I guess one of the things that she mentioned in the multiple choice section, of course, all the answers will be exact because, of course, it's multiple choice. You'll select the correct answer in the free response section. You will have a calculator. So it might be there might be times that it's appropriate to round your answer. If you do so, do so, of course, three decimal places. But your best bet is to write all answers exact as much as possible. You don't actually need a calculator in the free response section. There'll be very few topics that actually require the use of a calculator. There aren't any on this exam either. And so if you put all of your, if you didn't have a calculator, you could put all of your answers in exact form and that would be perfectly appropriate. But if you do want to round, round your answers to three decimal places. So let's look at the specific types of questions that you're going to see on this exam. One of the things I should mention is with regard to our lecture series, exam one covers lectures one through eight. Just as a reminder, lecture one introduced us to the Pythagorean equation, a review of triangles, distance, coordinate geometry and circles. Lecture two, continue on with this, talking about angles and triangles. Lecture three was about similar triangles, alternate to your angle theorem, among other things. That was chapter one. Chapter two, we started learning about right triangle trigonometry, solving story problems with triangles, reference angles, computing trigonometric ratios without a calculator. For some of those special angles, those were all in lectures four, five and six. And then lecture seven and eight for chapter three was circle trigonometry. We learned about radiant measure. We also learned about degree measure back in chapter one. We learned about how circles are involved in trigonometry and how they relate to triangles, sector, area, arc length, angular velocity. These are all topics that will be found somewhere on this exam.  You can find this, of course, on Canvas and solutions are attached to that. There are also solution videos you can find on Canvas if you want to see some more details on how you solve specific problems. What I'm going to do in this review video is not actually solve problems, but just tell you what type of question types you should expect. Because after all, this practice exam is only a sample of the types of questions you should see. There are question types that appeared in lectures one through eight that may not be appearing on this practice exam. They could be on the exam you take. So just because something's missing, doesn't mean it won't be on the test. And also just because something's here on the practice test doesn't mean it will necessarily be on the test. So this is just a sample to help you prepare. And I want to talk about specific question types. So in the multiple choice section, all questions will be worth five points. No calculator or notes will be allowed on this except for the formula sheet. Question number one, you'll be asked to compute a trigonometric ratio without a calculator. So you might be asked to compute sine, cosine, tangent, cotangent, secant or cosecant of an angle. And that angle will be one of our special angles. It will be something like maybe 30 degrees. So remember in the first quadrant, of course, you have these special angles here. Whoops, that doesn't look like a circle. So we have zero, of course, 30 degrees, 45 degrees, 60 degrees and 90 degrees. So in the first quadrant, you should, you will need to have memorized, of course, sine and cosine evaluated at these five angles. And because of sine and cosine, you can also compute tangent, cotangent, secant and cosecant given that those four functions can be written as fractions or reciprocals of sine and cosine in some capacity. So you should be able to compute sine, cosine, tangent, cotangent, cosecant and secant using the angle 0, 30, 45, 60 and 90 degrees. You can also compute a trigonometric ratio for any reference angle. That is to say, if you take an angle whose reference angle is 90, 60, 45, 30 or 0 degrees, that you should also be able to compute without a calculator. So if you got something like, oh, let's take tangent of 315 degrees, could you do something like that? Of course, because you could compute the reference angle of 315 degrees. You can use the fact that 315 degrees is in the second quadrant to determine whether it's positive or negative. And then you can compute the trigonometric ratio in that regard. You should be able to do this in degrees, but you should also be able to do this in radians as well. So could you do something like what's sine of pi over 2? And if you have to convert pi over 2 into 90 degrees, that's also very appropriate. I don't care. They'll be able to compute the six trig ratios for the five special angles in degrees or radians or any angle that references to the five special angles. That sounds like a lot, but this is very important, very fundamental stuff. Basically, if you were to take the unit circle diagram that we introduced, of course, earlier in this lecture series, lecture seven specifically, can you use that diagram to do a calculation without your calculator? That's the type of stuff we're asking about here. And so to be able to do this, if you need some more practice, please reference lecture six and lecture seven. Lecture six, of course, introduced to us the special angles 0, 30, 45, 60, and 90. Also reference angles were introduced in lecture six. And then lecture seven introduces those radians and the unit circle diagram. So that's the type of thing you're going to use on question number one. Question number two, can you convert an angle measurement from radians to degrees or can you convert an angle measurement from degrees to radians? So question number two is about conversion. If you want to convert from radians to degrees, what are you going to do? You're going to multiply whatever the given angle is by 180 degrees over pi. If you want to convert from degrees to radians, then you need to multiply the angle by pi over 180 degrees. And so that's what you're going to want to use in that situation. Angle conversion was talked about in lecture seven when we introduced radian measure. Question number three is going to be a topic coming from lecture one, which was just a review of very basic Cartesian geometry. Specifically, can you compute a distance between two points? The distance between two points? Let's say that p is given as x1, y1, and q is given as x2, y2. Then can you calculate the distance between p and q? This of course is the square root of x2 minus x1 squared plus y2 minus y1 squared as well. So use the distance formula. You might be asked to compute the midpoint of the two points p and q as well. And so of course the midpoint m is going to look like x1 plus x2 over 2 and y1 plus y2 over 2. So can you do these basic calculations of coordinate geometry that we saw in lecture one? Question number four is going to be sort of a continuation of that, but this one's going to focus on circles specifically. As we saw in chapter eight circles are a very important part of trigonometry. So you need to know the equation of a circle which is actually derived from the Pythagorean equation. So is the distance formula as well. And so remember the equation of a circle is given as x minus h squared plus y minus k squared is equal to r squared, where r is the radius of the circle which is half of the diameter and the coordinates h comma k are the coordinates of the center of the circle. So can you either create the equation of a circle satisfying certain conditions or can you read from the equation of a circle information about centers and radii and things like that? So moving down the page here, question number five will be a question to ask you to compute reference angles. So question number one may or may not ask you to compute a reference angle. Question number five will ask you to compute a reference angle and it will be and that's all it's going to ask you to do. So you'll be given an angle measure, like in this case you'll be given 195 degrees. You're asked to compute the angle measure, the reference angle, excuse me. The angle measure of course will always live in the first quadrant, so always be zero degrees to 90 degrees, but it could also be in radians. So this question could ask you to find the reference angle in radians, in which case the answer will then be in radians, so you need something between zero and pi halves. So reference angles that we mentioned before were introduced in lecture six, lecture seven introduces us to radian measure, so you were also asked in the homework to compute some reference angles of angles that weren't radian measure. Question number six is going to be a question about circles, also about circles, but it's going to be things like arc length. So you need to know the arc length formula s equals r theta. You might be asked to compute sector area, the area of a pizza slice, right? And so remember the area of a sector is equal to one half r squared theta. You might be asked to do something with angular velocity, which we've talked about some of those things as well. Angular velocity, remember that the velocity, the change of s with respect to time, is equal to r omega. We're omega here, remember v is the change of position with respect to time, and omega is the angular velocity, which is the change of theta with respect to time as well. Arc length and sector area was introduced in lecture seven. Angular velocity we talked about in lecture eight of course. When you were using these formulas arc length, sector area, or angular velocity here, these angle measurements must be in radians, must be in radians. If they're not in radians, then you will get the wrong answer. And as this is a multiple choice question, you can very likely anticipate that one of the distracting answers will be the number you get if you erroneously don't convert from degrees to angles, assuming that's a possibility. So be very cautious of that. Continuing on with the multiple choice section. Question number seven is going to ask questions about partner angles. What do we mean by that? So we talked about this in lectures two and lecture three. So what are some examples of partner angles? Things like vertical angles, which would be a picture like this. One and three are vertical angles that are going to be congruent to each other. You have supplementary angles when the two angles come together to form a line. Supplementary angles are important. You have complementary angles. What happens when you have two angles that come together to make a right angle? Complimentary angles as well. This picture, this diagram, this is the type of pictures we expect with like the alternate interior angle theorem. You have, for example, alternate interior angles, which are, of course, congruent. You are going to have corresponding angles. You're going to have consecutive interior angles. You have exterior consecutive interior, excuse me, consecutive exterior angles and what is it? Alternating exterior angles. Things like that. So you have these partner angle relationships that you're going to need to know. Of course, another one is like if you have a triangle with angles one, two, and three, the sum of the three angles always adds up to be 180 degrees. So use these partner angle properties to say stuff about what things are congruent, what are not congruent, and solve various problems related to these partner angles. All right, question number eight, which we can see here, this is a question coming from section eight, and this is going to be circle trigonometry, meaning that when you have the unit circle and you have a point on the unit circle P, x, y, you're going to be using the fact that the x-coordinate is cosine of the angle theta and y is the sign of angle theta, and then the other trig ratios are computed, of course, from their usual ratio situation. So understanding how trigonometry is defined using points on the unit circle, that's what you need for question number eight. Question number nine is going to be right triangle trigonometry. You'll be given the diagram of a right triangle, do not assume it's drawn to scale. You have a right triangle where angle is, the angle theta is indicated, but you do not have the measure, but you are still asked to compute things like sine, cosine, tangent, cotangent, secant, and cosecant, given that triangle. To make life easier, you'll probably be given a Pythagorean triple, like three, four, five, whatever, but you just want to know your basic so-catoa relationships. If you know so-catoa, you're going to be just fine on this question, just follow the information from the triangle, like so. Question number 10, you actually will be given an angle measure, but this is going to be one of our special triangles, which led to the special angles. So how do you use like a 30, 60, 90 triangle? How do you find out the missing information? Like this one's a 30, 60, 90 triangle. What about like a 45, 45, 90 triangle? How do you use that? So basically, you need to know sine and cosine of 30, 45, and 60 degrees, and you can do those various things. So like, for example, if you take the short side of a 30, 60, 90 triangle, the hypotenuse is double that. Can you do trigonometry in those basic situations, those I should say the special situations, 30, 45, and 60? That then brings us to the end of the multiple choice section. So we are now entering the free response section of the test. This part of the test will be taken separately from the multiple choice section. On this portion, you will be allowed a calculator and you are allowed a 3x5 note card. You won't have the formula sheet, but you can put whatever formula is on said formula sheet that you desire if you need those for these types of questions. There's going to be five questions in this section and on average, there are about 10 points each. Question number 11, you'll be asked to solve some type of triangle diagram. So like you see in this picture, there's two unknowns. There's this number H and there's this number X that we do not know, but we do know some things about distance. We do know some things about angle measure. So using right triangle trigonometry, maybe involving some circles too, but using right triangle trigonometry, find the missing unknowns like find X, find H. That's what you're asked to do right here. I do want to see the answers exact on this question. So do write this as like X is five times cosine of 17 degrees plus, you know, tangent of 50 degrees whatever it turns out to be. You will notice that in this section, the angles are not the special angles, 55 degrees, 62 degrees. You are not anticipated to know those trigonometric ratios. Give me the exact answers, but also give me a solution that's approximate to three decimal places. That would be very appropriate for question number 11 right here. And so you'll be asked to solve triangle diagrams. So triangle diagrams, we first saw in section four, lecture four, when we first introduced right triangle trigonometry, but we also more likely should look to question number five. Question number lecture number five, excuse me. Lecture number four did sort of very basic ones, very, very basic ones that was just basically a right triangle. Number five about story problems involving right triangles, those ones have these more interconnected diagrams and that's what you should anticipate on question number 11. It's also worth mentioning you can also talk about six special angles could come into play, but most likely you don't need any special angles on these type of questions. So moving on to question number 12, which is right here. Question number 12, you'll be given a diagram of the unit circle and you'll be given a picture looking very similar to this. You might actually get this exact same picture or you might get something that looks similar to it. So you have the unit circle you see right here, there's some other points like O, A, P, R, B, other labeled and again this is the unit circle. So what you're going to be asked to do is you'll be asked to prove some type of statement like this one, it asks you to prove that the line segment O, R right here is equal in length to secant of theta. So that's what you're asked to be able to do. This is something we did of course in lecture eight, we did things very, very similar to this. Prove this line segment is secant, prove this one's tangent, prove this one's cotangent. You'll be asked to do something like that and the basis of the argument comes from similar triangles. So from the diagram we should be able to extrapolate maybe two triangles that are similar to each other or something like that and use a similar triangle argument to do this. Similar triangles were discussed in detail in lecture three but lecture eight is the type of similar triangle argument you're going to do, something using the unit circle like you see right here. Question number 13 will be a story problem involving right triangle trigonometry. So some more of stuff like section five. Question number 11, there won't be a story problem. You'll be given a diagram and you're asked to find the missing pieces of the diagram. Question 13 though, no picture will be given to you. You'll be given the story like so. You are by all means encouraged, encouraged to draw a picture to not just to help you process it but also to help the greater know what you're thinking. But this answer, this question will have an answer, some type of numerical quantity. Give it to me in exact form and approximate it to three decimal places. That would be appropriate for this question number 13. All of these questions so far, let's see, question, I'll take that back, question 11 was worth 10, 13 was worth 10, but question 12 I believe was worth 8. So let's move down here. Question number 14, this will be another question involving sort of like partner angles and things like that. But unlike what we saw in the multiple choice section where we just had identified whether these angles were congruent or supplementary or something like that, this will ask us to solve for unknown quantities. So you see like there's a triangle right here with some angles we can solve for some of those angles. We have some supplementary angles, things like that. So use partner angle relationships to help you solve the unknown values x and y. So do a little bit of algebra based upon the geometric relationships. This is type of stuff we did in lectures two and three. That's where it was very, very heavy in our lecture series so far. Question number 15, hi Pac-Man, it's good to see you right here. But question number 15, this is going to be a story problem involving circles. So remember question 13 was a story problem involving right triangles. Question 15 will involve story problems of circles. So this will be things about area of circles or sector area of circles. That's what this question is asking. What's the area of Pac-Man that you see right here? That's kind of a fun question. You could talk about arc length. Like what's the circumference of Pac-Man? That'd be kind of a fun question as an alternative. Angular velocity, things like that. So those type of story problems we did with circles, particularly in chapter three, about lecture seven and eight, that's where we saw these type of questions. So be prepared to answer some type of story problem involving these properties about circles. And then question number 16 down here is just a reminder to turn in your no card. I do put your name on it. Sometimes the no cards get detached from the test by mistake. And so having your name on it is a good way to make sure you get the credit for making the no card, which you definitely do want to make a no card. And so that then gets us through the exam. I've illustrated the types of question types that you will see. If you want to, I would recommend if you haven't already, to take a look at the practice exam and try the problems out on your own. There are solutions attached to them so that you can check your answers or if you get stuck to get some help. But as you're studying for this test, please, please, please work with partners in the class, work with tutors in the tutoring center, include me, your instructor, you'll come by office hours and be emails, just ask me questions when appropriate. I'm here to help you as much as possible. Just let me know. And I think this test is going to be okay for you all. I'll see you next time, everyone. Bye.