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Activated carbon adsorption data for an organic chemical pollutant is presented below: Use linear transformation to produce regression coefiicients and the regressi...

Question

Activated carbon adsorption data for an organic chemical pollutant is presented below: Use linear transformation to produce regression coefiicients and the regression equation b. Plot the data along with the regression line Use the regression equation to predict the reaction Fale (~r) at the given concentrations Determine correlation coefficient Plot the data using equation 5 30 35 75 100 130 260 340 130 155 170 187 488 199 [92 230 [209 [ 209 250 280 290

Activated carbon adsorption data for an organic chemical pollutant is presented below: Use linear transformation to produce regression coefiicients and the regression equation b. Plot the data along with the regression line Use the regression equation to predict the reaction Fale (~r) at the given concentrations Determine correlation coefficient Plot the data using equation 5 30 35 75 100 130 260 340 130 155 170 187 488 199 [92 230 [209 [ 209 250 280 290



Answers

$(a)$ find the equation of the regression line, (b) construct a scatter plot of the data and draw the regression line, (c) construct a residual plot, and $(d)$ determine whether there are any patterns in the residual plot and explain what they suggest about the relationship between the variables. \begin{tabular}{|cc|c|c|c|c|c|c|c|c|} \hline$x$ & 38 & 34 & 40 & 46 & 43 & 48 & 60 & 55 & 52 \\ \hline$y$ & 24 & 22 & 27 & 32 & 30 & 31 & 27 & 26 & 28 \\ \hline \end{tabular} :

So I put all of my data in list one and list too. And then hit the linear regression button. And I found that the equation ended up being 3.912 Plus 1.71133 X. And we got a correlation coefficient which it doesn't ask for that. But it's important to look at a .9895. And then I did a stat plot graphing list one versus list too and having wise of one B. This regression equation. And when I look at that graph that graph looks very very linear. And the data points are not all specifically on the line. But when that line is drawn through them it looks very very close and we could tell that by that correlation coefficient that it was quite possible that it was going to look quite linear now for part C. After I've calculated that value for the regression equation, I went to list three and doctor instead of going through and finding the observed well we have the observed value but finding what the expected value was a predicted value by plugging all these list one values back into the equation and then subtracting. We do have that feature under second and list on A. T. I. 84 then the residual at the bottom. And so it's giving me all these residuals. When I look at my uh my chart For my residuals or my staff at it, the first residual comes out to be about .4. The next one is about .24. The next one is negative .58. So that gives you an indication of what the residuals should be. And now I'm going to go back to my staff plot And I'm going to turn my step plot on to go list one vs list three. So list one versus list three. And I'm also going to get rid of my regression equation as wise of one And just the second hair, list one vs list three and hit my wife someone and clear that out. And then again hit zoom number nine zoom number nine. And when I get that residual plot, that residual plot will always have that X axis in the center and my residuals, I have a result. You down here, I have up here, I have done here, down here, there's one up here and down here and I'm really not seeing it's kind of oscillating around this line. So I don't see not a not a particular pattern. There's no curvature in the data. So there's no pattern, which means that the linear model fits quite well, is quite good. Oh, no. All right. So I would use that model as a good predictor.

We can calculate the entropy of absorption using equilibrium, pressures and temperatures. The next thing that we need to do in this problem is a look at the table were given and say, Well, what equation will fit all the values that I need and that equation will look like this. We'll get that. The natural log of P one overpay to with respect to Fada is equivalent to the entropy of absorption multiplied by T two minus. He won, divided by r times t one times t two p is our equilibrium. Pressure teas, air temperature in Calvin and data is our surface coverage. Then what we need to do is look at the table, look at each column and then plug those values in to our equation. The first thing we'll do is we'll say, Consider the volume is point to pardon me. There should be in our right there. But the math remains the same. Well, say Consider the value of the volume to be point to. Then what we do is we look at the table and plug in the values. The natural log of 13 Pascal's over 80. Pascal's is equivalent to the entropy of absorption divided by 8.314 which is just the constant, are multiplied by this whole quantity. 308 minus 273. Calvin divided by 2 73 multiplied by 308. Calvin, when we simplify that will get negative. 1.82 is equivalent to the entropy of absorption. The thing that we don't know divided by R, 8.314 multiplied by 35. Calvin, divided by 2 73 multiplied by 308 and we do that will simplify our values and rearrange to get the entropy by itself will get that the entropy is negative. 36,351 0.9 jewels promote well. We could simplify that to make it a little bit of an easier value. And get that are entropy is negative 36.35 to kill it joules per mole. And finally we look at the table again and say, Let's look at the other column. So now we'll say that our volume is point for we'll get that The natural log of 27. Pascal's over 170. Pascal's is equivalent to the Anthill. P, divided by R 8.314 multiplied by 308 minus 273. Calvin, divided by 2 73 multiplied by 308. Kelvin again will simplify, and the right side of our create of our equation stays the same as before. But now our natural logs simplifies to negative 1.84 We'll simplify and rearrange. Get the entropy. By itself. The entropy will be negative. 36,751 0.3 jewels promote again. We can convert that into Kila jewels and say that if her volume was point for our entropy of absorption is negative. 36.751 Kill a Jal's Permal.

Okay for part a of this question, we're gonna be using a graft to graft this function. We can notice that it will look like us. Okay. And then use the regression model on your calculator to obtain why equals 8.9 to 35 Axe to the negative. 0.11228 to five. So this is part A, and we got this using the calculator. Okay, part be, we're gonna be using this. The model we just found to predict the value for X equals 9.2. So 8.9 to 35 times 9.2 to the negative. 0.1128 three fought to 3 8 to 5 equals approximately 6.3 onto part. See, we see that for each point, X comma y you can take the natural log to find the corresponding point. So I'm gonna make a chart over here. A little table, naturally. Axe, unnatural of Of why? Okay. 0.69 Approximately 2.1 approximately. And then we have one point 09 and 1.97 1.557 0.92 1.8 sex. You can see that we could make a scatter part of the data and we'll notice that they seem toe look like they're in a line. 34 If I were to connect these draw dotted line to show, you could see how the points look like their end a line approximately, which is what we're ending. Toe obtain. Okay, now we're onto Part D. That's a four part problem. We're using a linear regression model based on the data points that we got in part. See? And we can obtain a graph that looks something like Thus, we have four points. One, 223 four And the model. You should obtain your calculation. Be natural. Log of why equals negatives. Your 0.113121 natural of ax plus 2.9 131 This should be the linear regression model that we obtained

In this problem. We're using regression, and we're going to do almost the entire problem on the graphing calculator. So the first thing we want to do is grab a calculator and then go to stat and then go to edit. And we're going to type R X values into List one and R Y values into list, too. Now that we have all the data in the lists, our job is to find the power regression model. So we go to stat and then over to calculate and then go down until you find power regression. It's just under exponential regression. If you're using the T I a graphing calculator, press enter and then we're using list. One we're using was, too. You don't need to put anything in frequency list. And here in store regression equation, I'm going to store my equation in my y equals menu. And so if I go to Alfa Trace, that's a shortcut to find the Y one. Then select why one that tells the calculator where to store the equation and then we press enter and this is our regression equation. Now if we go to y equals well, see, that was pasted in. And just for the sake of writing down the answer, we can round those numbers a little bit and write that down. So that model was about why equals 2.75 x to the fifth in the next part of the problem. We're making a prediction. So if X is 7.1, let's find it's why Value. So I'm going to do this graphically, and I already have the function in y equals. And now I'm going to go to Zoom Stat, which is Zoom nine, because it creates a good window for a statistics graph. And I'm interested in what's happening when X is 7.1. So if I press trace and type in 7.1, well, I get a value that's off my screen. Apparently. So now I need to fiddle with that and see what's going on. Okay, so the problem is that to you, zoom stat, we need to turn on our scatter plot, so I'm going to go up to plot one. Turn it on. Now the calculator knows to you zoom stat with list one endless too Zoom nine and there's our power curve. So now if we press trace and make sure the cursor is on the curve and we type in 7.1. We can see the Y value there at the bottom of the screen. 49,616 0.3. All right, so we'll write down this answer when X is 7.1. Why is approximately 16 49,616 160.3. Now we're on to part. See, now we're converting our data from X and Y two natural log of X natural log of y. So we're going back to the calculator back to stat back to edit and then change the value. So instead of four, it's going to be natural law before you can just type that right in instead of 6.5 natural log, 6.5 et cetera. Keep doing that for all of the values. All right, so once those were all done, we have our new lists, and our goal is to show the scatter plot so we can go back to y equals plot. One is turned on already. So that's our scatter plot. I'm going to remove the y equals so that curve doesn't graph anymore. so we'll just clear that. And now back to zoom and back to Zoom nine for zoom stat. So what we've done is your natural logs to convert this from a power curve into a line. So here's our scatter plot. Next, What we want for Part D is the linear regression model. So the regression equation that fits this line So we go back to stat over to Cal HQ, and then we choose linear Regression, which is number four. Press enter. We don't need to store this one because it doesn't look like a rest too graphic or predict from it. And here's what we get. Why equals approximately five X plus one so well, jot that down. Okay, so our linear regression model was y equals five x plus one. And finally, for part E. We're doing sort of, ah, verification that this power model does relate to the linear regression model. And if you recall from your calculator screen when it found a linear regression model, it called the slope be are a excuse me. It called the Slope A and it called the Y intercept to be because the model was of the form y equals X plus be. And so what we're doing is we're taking our value of a five and substituting it in there. And we're taking our value of B one and substituting it in there and showing that we really do get the power model that we said we were going to get in the beginning. So if I do that, I have y equals e to the first times X to the fifth. Now each of the first is right around 2.7. And keep in mind that it's not truly exactly a one here. That was around ID number. So this is not truly e to the first. It's e to the something close to one, so it may be a little bit more than 2.7, and then we have X to the fifth. So how does that compare to the power model? We got for part? A pretty much the same thing


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