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A plant biologist conducted an experiment to compare the yieldsof 4 varieties of peanuts (A, B, C, D). A plot of land was dividedinto 16 subplots (4 rows and 4 colu...

Question

A plant biologist conducted an experiment to compare the yieldsof 4 varieties of peanuts (A, B, C, D). A plot of land was dividedinto 16 subplots (4 rows and 4 columns) The following latin squaredesign was run. The responses are given in the table to theright.

A plant biologist conducted an experiment to compare the yields of 4 varieties of peanuts (A, B, C, D). A plot of land was divided into 16 subplots (4 rows and 4 columns) The following latin square design was run. The responses are given in the table to the right.



Answers

An agriculture researcher wants to compare the yield of 5 corn varieties: A, B, C, D, and E. The field in which the experiment will be carried out increases in fertility from north to south. The researcher therefore divides the field into 25 plots of equal size, arranged in 5 east–west rows of 5 plots each, as shown in the diagram.

(a) Explain why a randomized block design would be better than a completely randomized design in this setting.
(b) Should the researcher use the rows or the columns of the field as blocks? Justify your answer.
(c) Use technology or Table D to carry out the random assignment required by your design. Explain your method clearly.

So in this question, we have Ah, at least square Regression Lane, given by y, is equal to X plus B, where A and B will be often from this given equation so on solving this equation will be quickly having bees equal to 19. Andi is a corrido for dinner we opened. Therefore, the least square regression line will be given otherwise equal to 14 x plus 19. Here It's given in the question that excerpt present in hundreds off pounds per acre, off fertilizer used and wider presents deal in bushels per acre. So after this in the second part, they're asking, We do confirmed the result in part a using regression feature of graphing utility. So that will be come from we can do it using craft. No. In the next part, I have to use the graphing utility to upload the data and the graph the linear model. So I'll graphic in the linear model. If I have this way, they will do for the next plus 19 it will be something like this. So this is X and this is way floating the point. Are we having one comma? 30 Do somewhere down, one going to do will be somewhere below the line. The next 0.1 point five comma 41 will be somewhere just above the line. 1.5 41 and then the third point to come a 48 that will also be somewhere above the line just to cover for did and the last 0.2 point 5 43 53. That will be below the line 2.5 53. So this is how the line looks like Onda. Now, in the last part, I have to predict the hell off fertilizer application off 1 £60 per acre. So in the last part, I have X equal to 1.6 that this 1.600 which is equal to 100 and 60. So corresponding to this exit by the substitute in the give any question that I reopen so right why is equal to 14 multiplied by 1.6 plus 19? So on simplifying this very welcome or Toby 41 point for So this unit I'll use this is 41.4 bushels per baker. So that's my answer. I hope it's clear. Thank you

So in this problem were given some data that were found on the relationship between fertilizer being put on some acreage and the yield of wheat on the acreage. Okay. And in the first part they've done the least squares regression for a line through the data of this form Y equals X plus B. And on that least squares regression we have come up with these formulas for A. And B. It was 1 74 and seven B plus 13.5 A. Equals 322. Okay so on this data then call this equation one in this equation too. I'm a little line just to separate my data here for a minute so I need to put these together so that one of the variables cancels out. Right well it might be easiest to go minus seven times equation one. So that's minus 28 B plus won't be plus will it? Will be minus 49 A. Equals 1 74 times negative seven. So that's a negative 12 18. And then do four Times equation too was 28 B plus 13.5 times four is 54 A equals 3 22 times four. That's 12 88. Add these together. The bees cancel out. So I have five A. He calls 70 don't I? Okay. So that means A. Is 14. All right. So then with A. Of 14 I can go back let's say into equation one into equation one. And so I have four B plus seven A. Well 14 times seven. Let's be seven times 14 equals 1 74. So that means four B plus 98. Of course 1 74. It's a track 98 from both sides. So four B is 76 and so B is 19. Okay. All right. That'll need her. Let's do that. We'll need her 19. So that means I have Y equals 14 X plus 19. Mhm. For my solution here. Okay, the next part says use regression feature of a graphic utility, confirm our results. So if I go over here to my graphic uh graphing calculator here already graphed the data for us in here as you can see. So then if I go why and I do approximately um A X plus B. Okay. I didn't really get much, did I? But if I go why one, that's what my data is and X one there like I should have done. Then I get the line going through the data. We see there and look at that parameters and be A. Is 14 and be as 19 with our square 2.98 This is a great fit for this data because the closer that are square it is to one the better our fit is for our line going through the data, at least amount of error. We have and look at that A and B. Are exactly the values that we found when we did the regression here. And so if we then graph the graph the data which we just did. And the linear model, which is what that line is is a linear model right from our data here, then it will look like that for us as well. Okay, so then part D. Says to use our linear model, perfect the yield the yield for fertilizer application of £160 her acre. So I didn't write that very neat. Let me let me clean that up per acre. All right, well the fertilizer in our table is in hundreds of pounds per acre, which means this is 1.6 or our ex going into our formula. And the yield here is why so, why? The yield equals 14 times 1.6 plus 19, and that is 1.6 times 14 last 19. That's 41.4. And the unit on this is bushels or acre from the information given in the problem and there we go.

Yes, problem. We're told that we have some, we're looking at basically we yield um and again, doesn't really tell us, and I guess it does yield in bushels per acre as a function of fertilizer in hundreds of pounds per acre. So as we did a bunch of test plots for test plots, so they put different amount of fertilizer. And we see we get different um yields. And unsurprisingly, you know, we see that the yields um yields go up as we increase the amount of fertilizer. Obviously, if we continue this to a lot more, we wouldn't expect we expect this to probably level off. Um But we can over this over this range here, we can see that these data points line falling a pretty close to a line. So we'll use a linear regression And how I got this using numbers on an iPad, it's all explained in problem 24. So here is our slope, so it's 13.8. And again, here's the function that I used to get it

Okay, So in this problem, it says, use the formula from problems 57 through 60 in order to solve it. And it gave us a table of values. Um, so we're using the formula N B plus Sigma X, which is the sum of the X values times, eh? He was of some of the Y values. Our second equation is that some of the exercise again times be plus the sum of the expert a square times, eh equals the sum of the product of the X and Y values. So I needed those all of those coefficients. So in in this case is for, uh, some of our X values. It's seven. There's some of our wives I use is 174. The sum of our ex square that is 13.5, and the some of the products excellent wine by is is 322. The next slide. I will set up those equations and go ahead and solve it. So we're gonna have four b plus seven a pickling 1 74 We're gonna have seven b was 13.5. Okay. Equaling 3 22 Okay, so to get my coefficients to line up a multiplying my first equation by seven. And my second equation before I give us 28 b is our coefficient for both equations. It's gonna change everything else, though. So for our first equation, able come out to the 49 or the coefficient will be 49 and 1 74 times seven is 12 18 second equation member 28 be So those will cancel four times 13.5 is gonna be 54 a and then four times 3 22 is 12 88. It's almost subtract my top equation from my bottom equation. That's gonna leave us with five a equaling 70 tried about five a Is 14 going back over here to plug in to the first equation I have for being seven times 14 is equaling 1 74 seven Fun times 14 98. So I'm just gonna go ahead and try it 98 from both sides. And so we have four being Is he going to 76 and dividing before I guess it's B because 19. Okay, so our final equation our line of best fit he's gonna be 14 x close 19 said it must know what would happen if we had 160 acres. So X was in terms of hundreds of acres. So it's asking. What it's asking me to do is plug in 1.6. So we're gonna have the amount of fertilizer needed for 160 acres. Would be why is 14 times 1.6 plus 19? Okay. And that's just gonna equal 41.4, Which makes sense because it fit pretty well on that chart. Okay, so thank you very much.


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