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Part ILinear RelationshipScenario: A pediatrician wants to determine the relation that may exist between a child'$ height and head circumference: She randomly ...

Question

Part ILinear RelationshipScenario: A pediatrician wants to determine the relation that may exist between a child'$ height and head circumference: She randomly selects eleven 3-vear old children from her practice, measures their heights and head circumference, and obtains the data shown below:Height (inches) 27.75 24.5 25.5 26 25 25.75 26.5 27 26.75 26.75 27.5Head Circumference(inches) 17.5 17.1 17.1 17. 16.9 17.6 17.3 17.5 17.3 17.5 17.5Ifthe pediatrician wants to use height to predict head

Part ILinear Relationship Scenario: A pediatrician wants to determine the relation that may exist between a child'$ height and head circumference: She randomly selects eleven 3-vear old children from her practice, measures their heights and head circumference, and obtains the data shown below: Height (inches) 27.75 24.5 25.5 26 25 25.75 26.5 27 26.75 26.75 27.5 Head Circumference(inches) 17.5 17.1 17.1 17. 16.9 17.6 17.3 17.5 17.3 17.5 17.5 Ifthe pediatrician wants to use height to predict head circumference, determine which variable is the explanatory variable and which is response variable_ Draw a scatter diagram of the data_ Draw the best fit line on the scatter diagram_ Does a linear relation exist between height and head circumference? How would you describe the relationship and why?



Answers

A pediatrician wants 3 to determine the relation that may exist between a child's height and head circumference. She randomly selects 11 three-year-old children from her practice, measures their height and head circumference, and obtains the data shown in the table. (a) If the pediatrician wants to use height to predict head circumference, determine which variable is the explanatory variable and which is the response variable. (b) Draw a scatter diagram. (c) Compute the linear correlation coefficient between the height and head circumference of a child. (d) Comment on the type of relation that appears to exist between the height and head circumference of a child on the basis of the scatter diagram and linear correlation coefficient. CAN'T COPY THE TABLE

So you're given a set of data that you have, what the height is of a child in inches, and you have what the circumference is of the skull and inches, and you should be putting all those values of your calculator into list one and list to in your calculator. And then they ask you to describe Our explanatory variable is given as the height, and we're letting the response variable be the height of the child that you're trying to determine what the head circumference is from from the height. So that's the response variable, and I'll just shorten up with that now. The next thing you're supposed to grab a plot and I have a plot graft on my calculator, and I have again a scrap scatter plot listed. I have the height of the child here and have the circumference of the head here, and when you grab this, it looks quite a bit like this grab. So it looks to be quite linear. And so then on Part C were asked to use and find what the correlation coefficient is. And so I use my Lin Lynn rag and you have a couple of Lynn rags on the calculator. The one you'll actually be using more often than not, is this one. This is the way Statistician's like it. And when you do that calculation, you find out the correlation coefficients 10.911 And so, um, that shows us that there is a positive association. So as the height increases, it appears as though the head circumference also increases, and it appears to be quite linear. So in answer to question D, it asked, Do we seem to have a linear relationship in this? This is yes. It's not perfect by any means. It does. That one line is not going to fit this perfectly. But since that correlation coefficient is quite close to one and we don't seem to have really any outliers on this, it seems to follow the pattern quite well. It definitely does look like a linear association. And then on Part E. Let's see what Part E. S. Part e asked us to convert the data over into centimeters so we could go over in our calculator and let me go up here and kind of erase this guy and you will be using your spreadsheet a lot on here and I can take list three, and I can have that be list one times 2.54 and then it will convert all those heights into, uh, into centimeters. And then I can go over to list four, and I can have that be list, too. Times 2.54 and again that will convert these head circumference measurements into centimeters. And then I again can do a linear regression between List three and list four. And what you'll find is that correlation coefficient will stay exactly the same. It doesn't care what unit you put your numbers in. That relationship between these, regardless of whether I have these in inches or centimeters, is going to create the same correlation coefficient.

It looks something like this. Now, using the linear regression program on your graphing calculator, you can graph the line of best fit, and it will appear on your graph that looks something like that. Now that we have the line of best fit. Let's use a graphing Catholics who write it down So we know the equation of the line of best fit, and we're going to write it and function notation. So function Notation says that if we have, we're using the height to find us a conference, as the conferences see, so we have C. So So we have see of age what age is the height and C is the circumference equals. And again I'm using the linear progression program on my graphing calculator. Ah, Brazil. To me, seventh day H, where H is the height plus 7.33 So that is the equation of the line of best fit. Now listen to her, but what the slope means. So here's the slope. The slope is still 0.373 on. Let's examine what happens when we increase age or the height by one. If we increase the height by one. The sea or the circumference of the head will increase by 0.373 So that is what the slope means for everything. One increase in H C increases by 6.373 So now that we have this equation, we can plug in a few points. So let's say that you have a height of 26. I think this isn't inches. Let's say we have a high of 26 inches and we want to find the head circumference of a child with a height of 26 inches. So we were plug and see of 26 equals 0.373 times 26 plus 7.33 Now against is we do have access to a calculator. I was to just just plugging this into your calculator because it gets a little nasty if you have to do this by hand and you end up with C being 17 0.3 So again, all we did is plug and 26 with H, and we got the corresponding sea or sycophants. Now we can do it the other way around. Let's say that we have us a conference of 17.4 inches, and we want to find the height of a child with a head circumference of 17.4 inches. So this time see equal 17.4. So we have 17.4 equals So 0.373 eight plus 7.33 again we have solving for H. So first thing that the do subtract 7.33 from both sides. And if you subtract 7.33 from both sides, you get 10 point 07 equals ill wind 373 eight. Now we want to sell for age. So divide both sides by 0.373 and we get H equals 20 seven. So a trial with a high of 27 inches will have a head circumference of about 70.

Here in party This cutter diagram like this here in part B using the Neemia Digression Moghaddam though line off best fit is see record at is equal to 01373 four That less 7.3 to 6 it Yet in part C for eat one inch increase in fight The circum friends increases by zero Porn 37 34 In here in part B see, record 26 is equal to zero Porn, please. On 34 Record 26 plus 7.3268 which is similar to 7.0 in just in part. E do find the height we saw the for lying. The question 17.4 is equal to 0.3734 at plus 7.3268 which gives 10.732 is equal to 0.3734 at no. Here comes the valley off edge, which is equal to 26.9. It it is similar to that with child the tile with me head circumference off 17.4 in just would have we Hi. Off they're old on this x 0.98 in just

All right, you guys. So for questions 17 here let's go and jump right on and I'm gonna go ahead and solve the entire problem using a google sheet and so let's go ahead and start there. That's starting with google. She does start a bit slower as you do have to input your own X and Y values. Let's go and do that. I'm going to take the X and Y values directly from the problem and we'll go from there. Mhm. So it does take an extra second but trust me it is worth it. Mhm. All right, so those are the heights and now let's see the circumference is right. Mhm. Mhm awesome. Okay, cool. And so now what we need to do is we need to part a which is to find the least squares regression line. Okay, so all I need to do here is very symbolizing to highlight the data and go insert chart and it does Perfect. Okay, it does give me a nice little scatter plot here. If it doesn't give you a scatter plot, not to worry. Um Yeah, it doesn't give you a scatter plot. Not to worry. It will you can make edits on that scatter plot with fairly basic ease here. Um Hold on I'm going to fix the window really quickly, it's not quite the size that I would like it to be perfect. Okay, cool. And so I'm gonna move my chart down here and actually make it a tad smaller as well. So I'm trying to fix my workspace a little here. Perfect. Okay, cool, so there is my chart. Okay so if so if you want to edit your chart first off, you just click on that the three dots hit edit chart. Um And sometimes it won't show up is scatter plot that shows it was like a hissed a gram or something or a column chart. If that happens, no big deal, just click on that, the type of chart and then just click on the scatter plot there. Okay um and before we're able to find the least squares regression line, the good thing is google sheets, do the whole thing for us. You do want to make sure that these two boxes down here and check they already checked for me. So that's good. The next thing we're gonna do is this I'm gonna go to customize and then I think it's series and I think that's right. Yes. Okay, serious. Customize serious. Okay and um and the series are going to scroll down just a touch and you're going to click on trend line. Okay. And there is the least squares regression line but we do need to find the equation as well. So there are equation is right is you can actually find the equation in the label. So the label right now it's not labeled as anything. You can label it with the equation itself and there is our equation. Very good. So that is um that's very very important that we do need to have that equation. Okay so that's good and that is um part a right there. I'm interpreting the slope and y intercept. So our slope means 0.183. So basically what that means is this when the height increases by one inch? Okay, so when the explanatory available variable increases by one, our explanatory available right now is height. So when our height increases by one inch interest, you need to measure that means our head circumference should increase by 0.183 uh inches as well. So for you will get a 0.183 increase in circumference for every increase, one inch increase in height. That is what the slope means. The why the intercept is actually in this case, the y intercept is not quite uh appropriate to interpret at this point because the y intercept occurs when our x value is zero. What it makes sense for a child to have a height of zero? No it does not does not make sense for that to happen. So we would actually say I'm interpreting in this scenario, interpreting the Y intercept would not be appropriate because you don't have a child with height zero. Um Okay so that's part of the part c Um we want to use the equation to predict for when a child is 25 in tall. Um so yeah so that's gonna be right over here. So I noticed we we already have 25 actually actually have a data point for 25 but let's go ahead and see um Our guest list we're going to make a prediction. Um So we're going to make a prediction here for 25 And let's see what happens here. So we're just gonna go ahead and uh input 25 more equations. So let's go and start typing our equation. I'm gonna start by doing equal sign in a new cell. I'm gonna go start typing in the equation and that X. Is right next to it. So I'm going to go times 25 because 25 is our input and then we're gonna add 12.5 And are predicted height is a touch over 17. Okay so we predict a touch over 17 and then this is um looks like this is going to be above average. How do I know that? Um This is gonna be uh oh I'm sorry I skipped part right there. Okay so that's part C. Right there. Part D. Is it wants us to compute the residual. So all we have to do is find the difference between um finally just between our actual value or predicted value. Okay so predicted value was this one. So I started by typing new cell equals um you forgot the labour residual. So this is our residual, it's a residual equals so type equals Um are predicted value minus our actual value and are residual is 0.175. And so um this is above average because that's positive. That is going to be above average. That takes care part d draw the least squares. Regression line. Got it. Um We already did that And then um no. Okay cool. And so then it wants us to notice this actually wants us, we have two instances where have 26.75. Um But even though they're both these two Children here are the exact same height. Um Their head circumference is slightly different. How can this be? Well it just it kind of makes sense right because not every person is exactly the same and that's possible because natural variable variability happens in datasets especially when it comes to human beings. So yeah that's just normal. I mean natural variability happens is what I might say. Um and then for part G what we would say is it would not be reasonable to predict the head circumference of a child who is 32" tall. Are data set works is strictly between 24.5 and 27.7 looks like 27.75. And so we do not want to predict data that is outside our data set because we cannot guarantee that the relationship will remain the same outside of our data set. So that would be inappropriate to predict um to use this regression line to predict Um the head circumference of a child who is 32" tall


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