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Elassic experiment by Dutton & Aron (1974) examined men ultracton t0 female they met in either one of two conditions: high unstable shaky bridge low sturdy bri...

Question

Elassic experiment by Dutton & Aron (1974) examined men ultracton t0 female they met in either one of two conditions: high unstable shaky bridge low sturdy bridge. Later, they rated the attractiveness of the woman The following data is generated replica of this study_ Attractiveness WS rated as 1-10 with 10 being very attractive. High Bridge Low BudgeSS] 201 6.82SS: = 154 M4.57The researchers want tO evaluate how the independent variable (bridge height) affects attractiveness ratings_ More

elassic experiment by Dutton & Aron (1974) examined men ultracton t0 female they met in either one of two conditions: high unstable shaky bridge low sturdy bridge. Later, they rated the attractiveness of the woman The following data is generated replica of this study_ Attractiveness WS rated as 1-10 with 10 being very attractive. High Bridge Low Budge SS] 201 6.82 SS: = 154 M4.57 The researchers want tO evaluate how the independent variable (bridge height) affects attractiveness ratings_ More specifically, they predict that those men in the higher bridge condition will rate the woman being more attractive_ State your null and alternative hypotheses. Whut= are your degrees of freedom and critical value(s) given an a 05 level? What is the estimated standard error? d.) Calculate your test statistic.



Answers

In $1990,$ the mean height of women 20 years of age or older was 63.7 inches based on data obtained from the Centers for Disease Control and Prevention's Advance Data Report, No. 347. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 63.9 inches. (a) State the appropriate null and alternative hypotheses to assess whether women are taller today. (b) Suppose the $P$ -value for this test is $0.35 .$ Explain what this value represents. (c) Write a conclusion for this hypothesis test assuming an $\alpha=0.10$ level of significance.

All right, So in this problem, we have Wei have data, pretend students. And for each student, they give us their own height and their ideal partners height. The first thing that has asked of us is to find a regression line on the correlation coefficient and car observations on it. Silk. Um, first, we're going to do that. A regression line. It looks something like this. It's, ah, equation for a line and due to find it. So if we need the value for M and B, that's the slope and the y intercept for the line. Um, to do that, we need to build this table. Over here. Come I used Excel Rivers on the first two columns that they we have here is the data that they give us this excellent light. Next we build the column that for each row they tell us the multiplication x y then iss X squared and y squared and the last hero we have because some of each Carlin So these values are the ones ever can use to find them and be so the's values I compute them over here and thes to equations are the definition for the Slope and slope and B. Why intercept of the regression line. So, um so our first thing to do is to plug in those values that we found on through the table. Um, and and the only one that's not here. And it's actually the number off, uh, the amount of data and we have We have ten entries and will be equal to ten. Okay, now, already so. And is ten. Uh, the some ex wise is forty six thousand, four hundred and ninety three. We said tracked first s o X. That is six hundred and eighty two times the sum of why that is also six hundred and eighty two. And that goes over, uh, n that's him. And next, we have the sum of the squares minus the square of the sun. So these were two different numbers. This sum of squares is going first, Queer X, and then we add all the the fact of Andries de So they're talking about this number right here. Thiss iss forty six thousand seven hundred and thirty. Huh? Uh, minus than this square. Uh, this number that's the summer exists. That is six hundred and eighty two and then this number he's cleared. So now we have this expression for them. It's all constants. So weakness. Either you do it by hand or use a calculator. I've found that is approximately equal to minus zero point zero night. All right, so that would be the slope. And now, to find the why intercept. Can you see the formula? Any definition, also from now on. So we stirred with self wise that it's six hundred and eighty two, six hundred eighty two minus, um And this is the one. The value that we just found that it's minus zero point. I scare nothing, turns the sum of excess POTUS also six hundred and eighty two, and that is over. And who has a value of ten Gandhi throughout constants. You can place it into the calculator and get seventy two. No, I'm sorry. Seventy four kind. Twenty eight. Keep. Okay. All right. So this will be the y intercept for the regression line. We can write out the equation for the line explicitly accusing this slope interests of former Emma. So we stopped. We found out this line. ISS What cause minus serial point. Syria, a nine thanks, plus seventy four turned twenty eight. And so this is good. Lined with the I'm sorry. The question for the line. Uh huh. So what's characteristic above this of the slope? Here it is. Almost zero. So that will be almost a horizontal line around this value. Okay, But we're not done. We are also asked to find the correlation coefficient, Uh, correlation coefficient, which we call our It is defined like this. And we're using the same values that we found on this table to calculate that so we can go ahead, answer plugging in these values. Uh, and it's tend the son of X y. It's forty six thousand four hundred ninety three good. Minus the sum of the outfits that is six hundred and eighty two times flies. That is also six hundred and eighty two. All this goes over over firstly, people, we start with square root. And that's ten again, where you were using Sum of squares and square Cem. So we begin with some squares. Hey, good. Education. Forty six thousand seven hundred and thirty minus minus. I said, uh, the sum of X, that is six hundred and eighty two. And this number is quick. That is our first school route. This is multiplied by another square room beginning. Also end. That's ten and some of squares minus the square of some. So ten times sum of squares for why isthe forty six thousand six hundred and seventy. Linus, Um, B x minus scripts. Um, so some is six hundred eighty two, and then this number squared so we can Oh, this is some new thinking square. Now we have this really big, but but just complete. No, it's all Constance. Which means we can either put that into the calculator, do it by hand, and I got that Our ISS negative. Zero point one two three. All right, So, um, so what if this number, this coefficient mean means first, our is closed to either one positive one or negative fun? That means that, uh, this line is a very good fit for the data. Yes, but if it is close to zero, it means it's not a very good fit. All of this not model. It is not a good model for the data, so this isthe very close to Ciro. It's definitely closer to Ciro them to one minus one. So what this is telling it. This line that's almost horizontal. It it's not a very good fit, huh? Model these trends for that. The ideal partner set. So that is a first for second part, huh? It is revealed to us that the first five data pairs are for female students and a second five for male students. So we're asked applying again. They're regression line on the correlation coefficient, but for each set of data separately. So we're going into the first fight first, uh, that isthe course months, two girls. So since we are doing the exact same thing to regression line and Correlation coefficient, we need a similar table. So now it's just the vice. But first, first five data points. Um, the approach is going to be the same. So we're not going to go into too much detail. They already know how to build this Temple Peace, bro. Oh, this really is the sum for each column. And then we're going to apply again. The for the definition. Put on your red definition for slough. Nope. And for eggs for slope. And why intercept and the court the coefficient. But worthies you've got so since We already did it once. I'm just gonna give you what I what? I thought so for slope. I got syrup point sixty seven, and with wind yourself got twenty seven point nine, we can write explicitly. The the regression, the question for the regression line. So M iss serial point sixty seven spoke close X. That's wind yourself. That is twenty seven point night. All right, so we have this new line that fits the data for the female students and the correlation coefficient can Using the definition is this number, and that is a lot of a lot closer to one. So that means this is a much better data for the The much better fete for the data. You see, the first back puts. All right, uh, now we're going to do the same thing for boys. We construct the people. This last role is this Last root is the use a little color? Uh, peace are the sums for each column, and these are the values were getting use to find the slope and intercept for the regression line. Um, I already use these definitions, and I found that the slope was sirrah point pretty tree and the one interested just thirty four point syrup. Or so we can provide the agression line that fits the the data with points the ISS. Why equals Sierra Point forty three times X plus? Why intercept That's twenty four points zero four and the correlation coefficient that we found us this. It's not as good as the one for the girls, but it's still a lot closer to one. So that means it's good enough fit for the data with Putz. So and that's part asked us to plot all the data and one graph using different types of points to distinguish the data for the males and the females? No. So, um, since I heard a lot of points already planted a few I'm using red for girls and green for voice. So we're missing just the last two points X represents. The height s is acceptable than white is the ideal partners. So good luck. It's less too. Censuses X will find it on this. Access seventy three will be around here and, of course, plunging y value. That would be the second coordinate is sixty six. It would be somewhere around here, so you go up go to the site and it will be somewhere around here. You. And for this, for the last point, seventy five here should be around up way between these two and sixty six. It should be on the same height. Cell, same Highness. This one a little bit to our right. There you go. Probably a lot less close. So, uh, so that would be Yeah. So here on the girl's data are over here. Um, voice data is a lot closer and under the gross data. So it as this It says that using this plant and the result of propriety explain why we found, in part a sow here for the girls. They have, ah, higher slope, Hunter Slope. And the white intercept is also lower. And here it's, uh well, it's slower at lower slope and higher Who? Why intercept? So this regression for the girls will be more unless around here, shit And for the boys, it'LL be around here. That would explain why the slope for the girls this a lot at a steeper but the voices throughout lower and the Y intercept for the grossest all the way over here. What? The voice is only, um, up here. Uh, so as further what we observed in print, eh? It was almost a core central line. So if we let's get these out of the way. If we were to take all of this into one data set the line, don't go through, it would be around here. It's kind of like averaging out both data sets. And that's why it would be almost home was the horizontal line. Hey.

However, one. So today we're doing a little bit more work on this state asset, which is about the height off women and specifically on female health professionals in America. And we did a little bit of work on confident falls in the last video using this data. And so now we're gonna be looking at testing the claim. That's the height of females in health. Fashion is different from the average height for women in the U. S. And so, just to remind you for a second off the values from this so this contains 50 values is contains 50 data points, and if you have them up on divide by 50 you're gonna find that the average is 64.7 eight. And we're told that the average height for females in the United States, which I'm gonna use me for is 63.7 inches, and the standard deviation of that is 2.75 Andi. So that should be all the information that we need to get working on. So to start with, we're gonna be testing a claim. So the first thing we need to do Yes. Oh, analyze first thing we need to do is we need to construct some hypotheses. So our hypothesis is that the average height off women, the health fashion is exactly the same is the average height of women in America. I said, that is 63.7. But I also I What's this is suggesting that the height of females in health freshen is different. And so that is saying that it is not equal to 63.7 on. This is important because this tells us we're doing a two title test. Yeah, And so next thing I want to do, let's calculate as itself all you So that star in a six X for which we have on the top left there minus mu all over Sigma, which is 2.75 I haven't reached 50. And if you put that into your calculator, you're gonna find that that comes out is 2.777 And so from this, we can find a P value. And so school just want sorry, really quickly. I've realized that I'm not quite clear on my notes whether I've were in the right thing down. So I'm using my country, you know, So we find out people using a calculator or using, um a the tables on you find that the p value for this I that stall value is, uh, no point. Not not 274 free. And so that's a p value on we're told to use the significance off No point nor five. However, remember earlier when I said it's really important that we know that this is two tailed. So that means we're testing both the upper and lower and and so what we can do we consult, approach this two ways, really? We can either say All right, Well, I want Alfa is gonna be no 20.0 to 5 on both of these ends. And then we compare this p value with no point nor to five. And what we find there is that this P value is significantly smaller. That so p is less than Alfa. So you reject ation all or alternatively, what you could do is you can say Okay, well, that's terrible, Curve. Let's try that again, is we can say OK, well, we're looking at both ends and I know that the area of this half is equal to my p value. And so, if I double my p value, I'm gonna get, um, not point. No, not ah. 5445 And then I compare that with the original Alfa, which is not quite nor five again. This should so show the exact same result, which is peace us an Alfa. So we reject. Okay, let me just make sure that that's enough. Enough for. And so that's two different ways of approaching a to tell. I thought sis, often I find this way easier because having what is usually a simple number isn't too difficult. Whereas doubling a longer decimal is a little bit harder. I suppose so. Either way, you're gonna get the exact same answer. And so what we find using this significance level is that we rejecting Haitian or so we're saying there's sufficient evidence using this sample that the high off females and health fashion is different from the average height of all females in the US So next part is a wordy question. It's suggest it's asking us how the rejection of the null hypothesis is revealed in the confident for that we came up with in the last video. So I'll just remind you of what that waas now. So the confidence before we came up with is this 65.54 So that's the continental we came up with on just to remind you again that the average height of women 63.7 sort of keep flicking up. And so we asked how we can use this confident fall to find that we would reject the null hypothesis. And essentially, what we said in the last video is that this value isn't in this interval on what that tells us is that we're certain within a 95% within about 95% that and we would reject the null hypothesis as they were actually doing the exact same thing by constructing a 95% confident interval were doing on then seeing whether this value is in that interval. We're doing the same thing if a little backwards to using a five point not five, um, level of significance in this test and then testing it that way. So the 95% confident Feliz the same is testing a 5% confidence in a test. I hope that makes some sense and specifically to reiterate the way that this reveals a rejection or hypothesis is because it does not contain this value that were given earthy national average. Then we're asked to explain how this interval could have been used to test the claim. Andi, that's exactly what I've just been saying. So we can construct a confidence interval like we did in the last video. And then we can essentially test whether the value that we're using in our hypotheses that we've got here is in the internal. And so what again that is essentially doing is exactly the same as this test. Just a little bit backwards. We would call this set of values the critical region. I'm not sure if this is technology you've come across before critical region, and what the critical region is is essentially the values for which you would reject. You would, um, reject the null hypothesis. And so not in this case is Excell Essence in before 0.2 four x greater than 65 0.5 full. And so those are the values that we would reject each not for on. So this value is in this critical region because it's less than 64.2 And so we reject a channel. And that's how we can use conference intervals to test the mean. And the final question is asking, How would we find a situation where we didn't reject the null hypothesis using a calm presentable and we wouldn't reject a the no hypothesis if the average was inside of this interval? If the so, if the actual average was between these two values, you know, if the population average, it would comparing again 64.2 then we wouldn't reject. And so that's how you would reveal no significant difference using a confidence interval.

In this problem, we're going to be testing the claim that the writ off left handedness among meals ISS less than that among females. And so we have two groups. Two samples and way have 240 males and 520 females. So out of the 240 males, 23 are left handed. So the proportion off left handed meals is 23 out of 240. And for the left 100 females, the proportion is 65 out of 520. And for this, uh, for this testing, we're going to use two approaches. The funds approaches the hypothesis test on the second approaches the confidence interval method and for the help. But this is test. We're going to use the 0.1 significant slept And since we are testing the clean that, uh, the rate off left handedness among males iss less than that among females, the now hypotheses will be P one is equal to p two. On the alternative hypotheses will be P one IHS less than p two, which means that the proportion off meals see meal is smaller than the proportion for females. And for that reason the critical value will be negative 2.33 Because this time we're saying that the proportion ISS less so since it's less the proportion of male is less than pushing off female. We use the negative, um, part off the pictographs. So negative 2.33 So now we want to get the calculated value of that. And to do so, we have substitute the values that we have here in the formula, and we obtain the value of that as negative 1.1 six. So now we can compare the critical value of that and the calculated value of that. So in this case, we see the critical region is the region to the left off negative 2.33 and the completed value of that He is not within the critical region. Uh, because negative 1.16 is to the right off the critical value. So this means that we'll have to make the conclusion to fail to reject denial hypothesis, which means that there is not sufficient evidence to support the claim that the writ off left handedness among males iss less than that among females. Now let's see how the the results would be for confidence interval test. So to test the claim, we would have to work out the value off e by substituting the values they into the formula. That is, the margin of error on the margin of error is 0.557 when you could substitute all the values correctly and also when you substitute the values correctly into this expression for the confidence interval, the thing you notice that the interval limits are negative. 0.8 47 Aunt Positive 0.267 So those are the confidence interval limits and this tells us that the difference between the proportions contains the difference. The confidence interval for this difference contains zero. It includes zero on this means that there does not appear to be a significant difference between the rich off left handedness, uh, among meals Andi riddle among females. Okay, so that doesn't appear to be a difference. And thanks. In other words, there is not sufficient evidence to support the claim that the rate off left handedness among males iss less than that among females. So in the last part of the question patsy. We're going to be using the results from the tool tests to tell if the writ of left handedness among males uh, it's less than the rich off left handedness among females. And we see that the two tests are in agreement that the rate of left handedness among this does not appear to be less than the rate off left handedness among females. Because we rejected the not we fail to reject the null hypotheses. Uh, then we can conclude that that does not appear to be any difference between the 22 categories the males and the females.

Let's look at this question. We want to construct a 99% confidence in trouble. And what is given to us is the observations are 78 to 10657 in 2106 five 7889597895978895 nine. How many observations are these? 123456789 10 11 12 13 13 and is equal to 13. So degree of freedom is 12. Degree off. Freedom is going to be 12. We're going to use a T distribution. Alfa a 0.1 AL 50.1 Hence Alfa by Do is 0.5 What is going to be s s is going to be again X minus X bar, whole square upon N minus one. This is my degrees of freedom. What we're doing here is basically trying to understand the way off doing this. Calculations are not important. You need to understand how to do this because all of these things are going to be done by the calculator Statistical software anyways, so right now we're just looking at the method. How do you construct a confidence interval expert? Plus minus T Alfa by two s by route. And this time over here is known as the margin of error. And after substituting the values here, how do we find the Alfa? By two? We already know the value of Alfa by two and we know the degree of freedom. We'll get the alphabet on. What is the interval that we're getting in this case? The interval that we get is 5.128 point 985.1 5.13 ft and sexually to 8.9. Eat. Okay, so what were the results? Tell us about the mean attractiveness reading made by the population. Well, this result tell says that we are 99 right? What is the confidence level? It is 99. So we are 99% sure that the two attractiveness level off the population lies between these two Wellings. Between these two extremes, this is our answer


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