Q1) Determine the parameters a and b so that f (x) = aebx fits the following data in the least-squares sense. 1.2 2.8 4.3 5.4 6.8 7.9 y 7.5 16.1 38.9 67.0 146.6 266.2
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- The following data were collected to determine therelationship between two processing variables and thehardness of a certain kind of steel: AnnealingHardness Copper content temperature(Rockwell 30-T) (percent) (degrees F)y x1 x278.9 0.02 1,00055.2 0.02 1,20080.9 0.10 1,00057.4 0.10 1,20085.3 0.18 1,00060.7 0.18 1,200Fit a plane by the method of least squares, and use it toestimate the average hardness of this kind of steel when the copper content is 0.14 percent and the annealing tem-perature is 1,100◦F.Use least-squares regression to fit a straight line y = f(x) = Ax+B to: xi -8 -3 0 4 6 yi 7 4 2 1 -2 Along with the slope (A) and intercept (B), A= ? B=?Determine the best (according to sum-of-squares-measure) curve y = Axb, through the dataabove.Transformed equation ln(y) = ln(A) + b ln(x)orY = a + bX.
- A consumer has $280 to spend on two commodities, the first of which costs $2 per unit and the second $5 per unit. Suppose that the utility derived by the consumer from ? units of the first commodity and ? units of the second is given by U(x, y) = 100x0.25 y0.75 How many units of each commodity should the consumer buy to maximize utility? Compute the Lagrange multiplier ? and interpret in economic terms.Find the point on the planez = x + y + 1closest to the point P = (1, 0, 0). Hint: Minimize the square of thedistance.It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let x be the number of fouls that were more than (i.e., over and above) the number of fouls made by the opposing team. Let y be the percentage of times the team with the larger number of fouls wins the game. x 1 4 5 6 y 51 41 33 26 Find x.x = Find y (in %).y = %Find the equation of the least-squares line ŷ = a + bx. (Round your numerical values to four decimal places.)ŷ = Find the value of the coefficient of determination r2. (Round your answer to four decimal places.)r2 = What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? (Round your answer to two decimal places.) %What percentage is unexplained? (Round your answer to two decimal places.) %(e)If a team had x = 3 fouls over and above the opposing team, what does the least-squares equation forecast for y (in %)? (Round your answer to…
- . Show that the estimate and the estimation error for an MMSE estimator are orthogonal (Hint: X and Y are orthogonal if E[XY ] = 0).Use the method of variation of parameters to find a particular solution of the DE…The data below were collected by a chemical engineer. Y is filtration time, x_1 is temperature, and x_2 is pressure. Fit a second order model. What operating conditions would you recommend if the objective is to minimize the filtration time? x_1 x_2 y -1 -1 54 -1 1 45 1 -1 32 1 1 47 -1.414213562 0 50 1.414213562 0 53 0 -1.414213562 47 0 1.414213562 51 0 0 41 0 0 39 0 0 44 0 0 42 0 0 40
- Determine the best (according to sum-of-squares-measure) curve y = Axb, through the data above. Transformed equation ln(y) = ln(A) + b ln(x) or Y = a + bX.The following data shows the atmospheric pollutants yi(relative to an EPA standard) at half hour interval xi. Find the equation y=a+bx of the least square line that best fits the data points given by 2,1, 5,2, 7,3, 8,3. Hence predict the atmospheric pollutant at x=6 half hour.determine the equation of the orthogonal trajectories of the following families curves: The cissoids, y^2=x^3/(a-x)