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The yield of a chemical process is related to the con- centration of the reactant and the operating temperature. An experiment has been conducted with the following...

Question

The yield of a chemical process is related to the con- centration of the reactant and the operating temperature. An experiment has been conducted with the following resultsYieldConcentrationTemperature 150 180 150 180 150 180 150 18081 89 83 91 79 87 84 901.00 1.00 2.00 2.00 1.00 1.00 2.00 2.00CamscannerFind the regression model and comment on its significance

The yield of a chemical process is related to the con- centration of the reactant and the operating temperature. An experiment has been conducted with the following results Yield Concentration Temperature 150 180 150 180 150 180 150 180 81 89 83 91 79 87 84 90 1.00 1.00 2.00 2.00 1.00 1.00 2.00 2.00 Camscanner Find the regression model and comment on its significance



Answers

Data Analysis An agricultural scientist used four
test plots to determine the relationship between wheat
yield $y$ (in bushels per acre) and the amount of fertilizer
$x$ (in hundreds of pounds per acre). The table shows the
results.
$$
\begin{array}{|c|c|}\hline \text { Fertilizer, } x & {\text { Yield, } y} \\ \hline 1.0 & {32} \\ \hline 1.5 & {41} \\ \hline 2.0 & {48} \\ \hline 2.5 & {53} \\ \hline\end{array}
$$
(a) Find the least squares regression line $y=a x+b$
for the data by solving the system for $a$ and $b .$
$$
\left\{\begin{array}{l}{4 b+7.0 a=174} \\ {7 b+13.5 a=322}\end{array}\right.
$$
(b) Use the linear model from part (a) to estimate
the yield for a fertilizer application of 160 pounds
per acre.

So this is the information. We're given the problem. We have a data set with three treatment groups 123 And in each treatment group, there are five elements. So we have a total end of 15 and we have three treatment groups. So the first thing we have to find in order to do anything with the Nova is the mean of each treatment group. So this is our noble table. This is what we're trying to find out the very end. Um, before we can do that, we have to find the mean of each treatment group. So our first mean is going to be the sum of each individual element. Um, under the 50 degree column, so would be 34 plus 24 plus 36 plus 39 plus 32 divided by the number of elements in that column, which is five. So we have a we get a value in the end of 33 and then for the mean population where the temperature is 60 forget 29. And then when the temperature is equal to 70 we get a means of 28. Now we have to calculate a grand mean, which we will do no as x double bar, which is equal to the sum of each of these individual means over the number of treatment groups. So we will get 33 plus 29 plus 28 over three, just a value of 30. So this is our X double bar. This is our grand mean, This is our grand mean, and now we have to find the sum of squares for our treatment. And the sum of squares for a treatment follows the this formula which basically says that you were going to take the difference between each individual item in each treatment group minus, um the the grand mean sorry that just say grand mean minus the grand means square multiplied by the number of elements in each group. So to specify, we're going to take, um five because that is the number of elements in our first treatment group times the first sample mean, which is 33 minus our grand mean of 30 squared plus five, which is the number of elements in our second treatment group, minus times 29 minus 30 squared plus five times 28 minus 30 squared and then we get a final value of 70. So this is the sum of squares of our treatment group. So we get a value of 70. Let me do that a different color of 70. And now we have to find the sum of squares for our total. And, um uh I felt like the formula was a little bit confusing for me, so I like to write it out in words. So we're going to take some of the difference between each element and the grand means squared. So we're going to take the difference between 34 our grand mean of 30 and then square that and add it to 24 minus 30 squared. Um, plus 36 minus 30 squared. So let me write that out. So this is equal to 34 minus 30 squared plus 24 minus 30 squared plus 36 minus 30 squared. And then we're going to do that all the way till the very end of our data set until we get to 31 minus 30 squared all the way at the end over here, here. And then we get a sum of squares of our total equaling 306 sum of squares were total is 306 and now we have to find the sum of squares for error. And it is simply the sum of squares of our total minuses on the squares or treatment which is equal to 306 minus 70 which is to 36. Now we have defying the mean square for our treatment and the mean square for error. And in doing so, we will discover the degrees of freedom for our treatment and degrees of freedom for our error. So first thing we'll find is a mean square of our treatment, which is simply the sum of squares of our treatment over the degrees of freedom for a tree degree. The degrees of freedom for our treatment is the number of treatment groups are K minus one. So this is equal to three minus one, which is equal to two. So we have a sum of squares of our treatment of 70 over to, and we get a value of 35 so 35 goes here and then we have a degrees of freedom of too, because that is our K minus one. Now we have to find the main square of our error. It is simply the sum of squares of our air over the degrees of freedom of our error. The degrees of freedom of our error is equal to the total number of their elements that we have. We have 15 elements. That is the end of our total minus R K number of treatment groups that we have, which is three. So we get a degrees of freedom of 12. So we have also a sum of squares of our error of 236. So it's going to be 2 30 six. Divided by 12 is equal to approximately 19 point six seven 19.67 And we also discovered that our degrees of freedom was 12. And now we have to find a degrees of freedom for our total degrees of freedom for our total, simply the end of our total. The total number of elements that we have minus one. So we have 15 elements in total minus one sequel to 14 Now, with the last thing we have to do to fill out or a nova table. Let's find an F statistic. The F statistic is simply the means square for treatment over the main square of our air, which is 35 over 19.67 which is approximately 1.78 And now, using this, we will get a P value. Um, again, I like to use Excel, so, um and except we're going to write the for following formula and some cell. Um, first we're gonna put in our F statistic. Then we're gonna put in the degrees of freedom for our treatment, which was two and the degrees of freedom for our error, which was 12. We got a value of 0.2104 and in the very end, it asks us to, um, use a 0.5 significant level. So Alpha's gonna be 0.5 and compare our P value, which we just discovered to be 0.2104 So that's your pee is point to 104 So because our P value is greater than our doubtful value, we do not have sufficient evidence to reject it all. But what exactly is the no hypothesis in this case, Tom, Because we're doing in a nova the null hypothesis is that the population means for all our populations are equal, that there is no variance in our population means. And our alternative hypothesis is that, um the means for the three treatments are not all equal. The three treatments O r not o equal. So because our we do not have sufficient evidence to reject the knoll, as we just discovered here, um, it means we don't have sufficient evidence to support the claim that the means for three treatments are not equal.

So if you're given like a hypothetical reaction. Um and you have the amount of a. Let's put coefficients. You have the amount of of reacting A. Um that is available to react. And you know the actual amount of B that's produced. The person field would be the amount of B. That's actually formed over. Um The theoretical amount of the and what is the theoretical amount theoretical? Let me just write it first theoretical Amount would be and we multiply it by 100% because it's a percentage yield. So theoretical amount is if you know how much a. You have using strike geometry and the mole ratios, how much be could possibly be formed. Now if you use the moles of actual to theoretical yield, you have like you know some amount okay of moles of be over some other amount of mousavi. And then if you wanted to um find the the same percent yield in grams, you would be multiplying by um the molar mass of B. And then by the molar mass of be on both sides, top and the bottom. So the value of the fraction would not change except within like the magnitude of era of rounding in um where you run off the moles and where you run off the molar mass. Um But the ratios should be the same value for percent yield in moles and mass because of the fact that you're multiplying the top and the bottom by the same number to relate the two.


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