4

For A =and b =a least-squares solution of Ax = b is X = Compute the least-squares error associated with this -3solution.The least-squares error is (Simplify your an...

Question

For A =and b =a least-squares solution of Ax = b is X = Compute the least-squares error associated with this -3solution.The least-squares error is (Simplify your answer: Type an exact answer; using radicals as needed )

For A = and b = a least-squares solution of Ax = b is X = Compute the least-squares error associated with this -3 solution. The least-squares error is (Simplify your answer: Type an exact answer; using radicals as needed )



Answers

(a) find the least squares regression line and (b) calculate $S$, the sum of the squared errors. Use the regression capabilities of a graphing utility or a spreadsheet to verify your results. (GRAPH NOT COPY)

We have these four data points 04131120 And we want to find the line that goes through those that best goes through those data points. Who best fits the data points? So a linear model. So we construct our sum of squared errors. This data point gives us b minus four squared. This one gives us a plus b minus three squared. This one gives us a plus b minus one square and this gives us to a plus b squared. So we want to minimize this so we can take the partials, partial derivatives. But the A and B. And said to me, go to zero and let's see here, we get um I didn't go to all the details here, but I'm not sure that you can do that departure with respect to A is 12 A A plus eight B minus eight and that has to be zero. And then the partial respect to be is eight times A plus B minus two. That has to be or zero. So we have two equations into unknowns and we can solve those. And it turns out that then A is minus two and B is four. And if we plug back into here we get that s across to so again, not zero. Because as you can see here from our plot, this line does not go through the data points. Exactly, it does go through let's see, it goes through this one, exactly, and doesn't go through this one. Exactly, I think it does. Let's see your line is minus so X equals two. Yeah, it goes through this one exactly, but obviously it misses, it can't figure out what to do with these two. And so it actually basically split the difference right here. So when it winds up being too here, instead of, you know, can't decide giving it to points one and three. And it basically just takes the average of those. So, you know, we get quite a bit of error about these points. So, you know, either this is there's some noise in this data or this just isn't a good model. And obviously it's kind of funny to have two different values for for one X value, whatever it is. And so if we took a measurement basically, you're basically saying that you have some, you know, noise in your data that you would expect that this wouldn't change or maybe there's actually another variable that's changing as in addition to X that you're not accounting for. So anyway, that's the best fit line minus two X plus four.

Mhm. In this case were given, let's see here eight points and I plotted them all here. So again we can see that clearly is not going to be a line that fits through the exactly through here. Um basically have this step kind of thing, but we can find the best approximation that we can um using uh minimizing the sum of the squared errors and I didn't write that all out because again that would just be really long. So generally we have a summation of all of our data points of eight times the x minus plus B minus the y value for each of these um eight data points. Now, if we take the partial with respect to those and we can again do that um it's just, you know, it's not a hard calculus problem but we can take the partial derivative and after some simplification we get to 32 A plus 56 B minus 74. And that needs to be zero for this to be a minimum and take the partial respect to be, and we get eight times the quantity seven A plus two, B minus two equals zero. And so that has to be zero for this to be minimized. And so that gives us two equations for two unknowns. So we can then go through and solve those And turns out we get some fairly even numbers a is 1/2 and B is -3/4. Now, we can then plug those back into here and calculate what S. Is and we get S equals three halves. So are putting it into the spreadsheet. Um It gave us a value for the linear regression Of a equals 1/2 and b equals -3/4, just like we had here. And again, S. Is not zero and that's clearly the case because in fact it doesn't go through any of these data points, but it does its best to fit through the the average of them. So again, this clearly looks like the best fit line that we could possibly have to fit this data.

In this problem, we want to use differentials to approximate the maximum possible air and each computed value given that X and Y have errors that are at most are percent and s percent. So to get started, we will take the differential of each function the absolute value of its differential over the function itself. So for part A, we get that this is why d X plus x d y all over. X y part B is the absolute value of one over why the X minus x over y squared de y all over x over y part C is the absolute value of two x times y cube DX plus three X squared times. Why squared, do you? Why all over x squared y cube and then in party we get that this is three x squared times y to the one half DX plus one half x cubed times. Why, to the minus one have do you? Why all over x cubed times y to the one half and then from here in part a, we get that this is the absolute value of D. X over X plus de y over y and in part B. We get that This is D X over X minus de y over y in part C. This is the absolute value of two times DX over X plus three times. Do you? Why, Over why and in Part D. This is the absolute value of three times DX over X plus one half times d y over y. And then from here we can apply the triangle inequality to all these cases. So in part A, this is absolute value of D X over X plus the absolute value of D Y over y and part B. This is less center equal to the absolute value of the X over X plus absolute value. D y over y in part c, we get that this is less than or equal to two times absolute value of D X over X plus three times absolute value of D y over y and then in part D. We get that this is less than or equal to three times the absolute value of D. X over X plus one half. We get that this is three times absolute value of D. X over X plus one half times absolute value of D Y over y, and we will use these to approximate the maximum the maximum possible errors in each in each function in each computed value. And we know that the heirs that X and Y have errors that are at Mostar percent and s percent. So in part, A, this is less than or equal to are over 100 plus s over 100 and this is less than or equal to, and part B are over 100 plus s over 100 and in part C, we get that this is less than or equal to two times are over 100 plus three times s over 100 and then in part D, we get that This is less than or equal to three times are over 100 plus one half times s over 100. So in terms of percentages, part A is R plus s percent, Part B is the same. And then for part C and part d. All right, those right here. All right, come under under the work, So it's easier to see, and that completes the problem.


Similar Solved Questions

5 answers
Calculate the volume of the solid of revolution formed by rotating the region bounded by y=r -1,x-axis: Y - axis. in the fourth quadrant about the y - axis:InxFind the length of the arc given byfrom x=lX=2Find the surface area formed by revolving the arc given by y =from X=0X=2 about the X-axis_
Calculate the volume of the solid of revolution formed by rotating the region bounded by y=r -1,x-axis: Y - axis. in the fourth quadrant about the y - axis: Inx Find the length of the arc given by from x=l X=2 Find the surface area formed by revolving the arc given by y = from X=0 X=2 about the X-...
5 answers
A solution of [Fe(bipy):]? was exposed t0 light at 512 nm in a spectrophotometer and gave an absorbance reading of 0.42_ The molar absorption coefficient, €,at 512 nm, is 7.0 x 10" Lmol.cm Calculate the concentration of [Fe(bipy)a]?
A solution of [Fe(bipy):]? was exposed t0 light at 512 nm in a spectrophotometer and gave an absorbance reading of 0.42_ The molar absorption coefficient, €,at 512 nm, is 7.0 x 10" Lmol.cm Calculate the concentration of [Fe(bipy)a]?...
5 answers
W hat are +he Dimensic-s Rec tagl e U +o Perimeter 12 Ona Dias Oko 20
W hat are +he Dimensic-s Rec tagl e U +o Perimeter 12 Ona Dias Oko 20...
5 answers
Part AConsider the reaction2H;PO4-P2Os + 3H20 Using the information in the_following_table calculate the average rate of formation of PzOs between 10.0 and 40.0 s. Time (s) 10.0 20.0 30.0 40.0 50.0 [P2Os] (M) 2.80x10-3 |5.80x10-3 |7.6ox10-3 .80x10-3 40x10-3 Express your answer with the appropriate units.View Available Hint(s)pARate of formation of PzO5ValueUnitsSubmit
Part A Consider the reaction 2H;PO4-P2Os + 3H20 Using the information in the_following_table calculate the average rate of formation of PzOs between 10.0 and 40.0 s. Time (s) 10.0 20.0 30.0 40.0 50.0 [P2Os] (M) 2.80x10-3 |5.80x10-3 |7.6ox10-3 .80x10-3 40x10-3 Express your answer with the appropriate...
5 answers
IGuhalGaldeeodr[nn Ordn &urititau BJak CankAl DadatC244Anahe BlentEDAEueeUant tht I ethisu about te |utate Vnt TWuu Feunte #valeree cachrcra U7 u (Kal_t4TTeaee neIteze 04 Let4 4Daruu JhcWteMctW BtokTo738
IGuhal Galdeeodr [nn Ordn &urititau BJak CankAl Dadat C244 Anahe Blent EDAEuee Uant tht I ethisu about te |utate Vnt TWuu Feunte #valeree cachrcra U7 u ( Kal_t4 TTeaee ne Iteze 04 Let4 4 Daruu Jhc WteMct W BtokTo 738...
5 answers
0! Wl acetate 5.65 5.67 4.67 3.67 has has made made (1 21 point) 'A4d point) of "59*5 mixing 4.67 mixing hao What 1SmL of Ohepk the 0.10M pKa pKa acetic of acetic the acid . the acid? acid? acid and and 20 15 mL mL 1 1 sodium sodium
0! Wl acetate 5.65 5.67 4.67 3.67 has has made made (1 21 point) 'A4d point) of "59*5 mixing 4.67 mixing hao What 1SmL of Ohepk the 0.10M pKa pKa acetic of acetic the acid . the acid? acid? acid and and 20 15 mL mL 1 1 sodium sodium...
5 answers
Based on the titration curve t0 the right for generic basic salt a) List the dissolved species as they exist in solution at each of the following points:Species from Species from Point the unknown the titrant Vu Maea Mhe LB 4Ho Wvu Le ,#} HEHR a 4e Vat Hc What is Kal for the conjugate diprotic weak acid? PH . pldbut /9 27) ~Philal] Lc) k6 _pH [E92L%100 ,Answer:loKa K' Il . IcmL of 4.00 M HCI added
Based on the titration curve t0 the right for generic basic salt a) List the dissolved species as they exist in solution at each of the following points: Species from Species from Point the unknown the titrant Vu Maea Mhe LB 4Ho Wvu Le ,#} HEHR a 4e Vat Hc What is Kal for the conjugate diprotic weak...
5 answers
Ana 4 + &+c ucl - [email protected] Uc wa (rey Lf Solve pc 6 4-4 2 = (~4 5 ) wler a = ( 5) m etlat + Elimth ~on lt Gau ss/an lb) ULS 2 296a (ovP Splve X s#knat 3X-49 + 4 ~ 712 2X -y +62 LLDI Ret
Ana 4 + &+c ucl - [email protected] Uc wa (rey Lf Solve pc 6 4-4 2 = (~4 5 ) wler a = ( 5) m etlat + Elimth ~on lt Gau ss/an lb) ULS 2 296a (ovP Splve X s#knat 3X-49 + 4 ~ 712 2X -y +62 LLDI Ret...
1 answers
Solve the equation_ You wlll need to use the factoring techniques that we discussed throughout this chapter. 8)(*(smaller value) (larger value)
Solve the equation_ You wlll need to use the factoring techniques that we discussed throughout this chapter. 8)(* (smaller value) (larger value)...
5 answers
Using shape representations, draw the combinations of atomic orbitals which form each of the following molecular orbitals in Iz (as labelled on the diagram): i) 01 ii) 62 iii) 63 and iv) TlaMolecular Orbital Energy Level Diagram for lodine, I2Tza T2bnpnpTla Tlb0}
Using shape representations, draw the combinations of atomic orbitals which form each of the following molecular orbitals in Iz (as labelled on the diagram): i) 01 ii) 62 iii) 63 and iv) Tla Molecular Orbital Energy Level Diagram for lodine, I2 Tza T2b np np Tla Tlb 0}...
4 answers
Problem 4(25 points) Suppose andand the angle between and W is 8 FindIa] tan €(b) 0Note: You can eam parial credit on this problar:
Problem 4 (25 points) Suppose and and the angle between and W is 8 Find Ia] tan € (b) 0 Note: You can eam parial credit on this problar:...
2 answers
Consider continuous-time Markov chain with a state space {1,2,3} with A1 = 2, A2 = 3, A3 = 4 The underlying discrete transition probabilities are given by 0 0.5 0.5 P = 3 8 0.5 0.5a) Find the generator matrix: (b) Find the stationary distribution of this CTMC.
Consider continuous-time Markov chain with a state space {1,2,3} with A1 = 2, A2 = 3, A3 = 4 The underlying discrete transition probabilities are given by 0 0.5 0.5 P = 3 8 0.5 0.5 a) Find the generator matrix: (b) Find the stationary distribution of this CTMC....
3 answers
18. Let P be as in Exercise 1 7. Provethat P2 js the principal ideal (2). Let P be the ideal {2a + (1 + VSbla,beZIv-J} inZIV-5]: Prove that r+SV-SePif and only if r= $ (mod 2) (that is rand $ are both even or both odd)
18. Let P be as in Exercise 1 7. Provethat P2 js the principal ideal (2). Let P be the ideal {2a + (1 + VSbla,beZIv-J} inZIV-5]: Prove that r+SV-SePif and only if r= $ (mod 2) (that is rand $ are both even or both odd)...
5 answers
QurnonFuy: 6ae n{ Mue Ofta peuple whuee Uave |n eueabtneeila Hautuudquestion 8Thc finishing process new furuIE Lenvch Jight blcmahc The tble bclow duplayt in the finish of new fumitute-Number e Blemisnes Probabillty0 340,250 190,110.07how many delects #ould WC ctncci On picce of furniture? On averuge, 0.28 0.85 0 25
Qurnon Fuy: 6ae n{ Mue Ofta peuple whuee Uave |n eueabtneeila Hautuud question 8 Thc finishing process new furuIE Lenvch Jight blcmahc The tble bclow duplayt in the finish of new fumitute- Number e Blemisnes Probabillty 0 34 0,25 0 19 0,11 0.07 how many delects #ould WC ctncci On picce of furniture...
5 answers
-14 POINIS L SCALCET8 1(c) Using 3 RHelp?1Mliderorol 6'
-14 POINIS L SCALCET8 1 (c) Using 3 R Help? 1 Mliderorol 6'...
5 answers
Cul3pbQuestion 4spherical mirrors % statements about Which of the following correct? real Image always produces A concave mirror Minujl imjge produces convex mirror always virtual lmace produces concave mirror alwjys produces real imaEC convex minor always
cul 3pb Question 4 spherical mirrors % statements about Which of the following correct? real Image always produces A concave mirror Minujl imjge produces convex mirror always virtual lmace produces concave mirror alwjys produces real imaEC convex minor always...
5 answers
What is the major organic product of the following reaction?Warin
What is the major organic product of the following reaction? Warin...

-- 0.020143--