Number and Algebra
Functions
Geometry and Trigonometry
Statistics and Probability
Calculus
Find the cubic polynomial that fits (0,0)(0,0)(0,0), (1,1)(1,1)(1,1), (2,8)(2,8)(2,8), (3,27)(3,27)(3,27) exactly, and comment on its significance.
For the points (1,3)(1,3)(1,3), (2,11)(2,11)(2,11), (3,33)(3,33)(3,33) and (4,83)(4,83)(4,83), verify whether they lie exactly on the cubic model y=2x3−x2+x+1y = 2x^3 - x^2 + x + 1y=2x3−x2+x+1.
Determine the residual sum of squares (RSS) for the cubic model y=x3−2x2+x+1y = x^3 -2x^2 + x +1y=x3−2x2+x+1 on data (0,1)(0,1)(0,1), (1,1)(1,1)(1,1), (2,5)(2,5)(2,5), (3,19)(3,19)(3,19).
Suppose a dataset (x,y)(x,y)(x,y) with x=1,2,4,5x=1,2,4,5x=1,2,4,5 and y=3,17,129,314y=3,17,129,314y=3,17,129,314 is said to follow y=ax3y=ax^3y=ax3. Determine aaa and check the fit.
For the model y=2x3−x2+3x−5y = 2x^3 - x^2 + 3x - 5y=2x3−x2+3x−5, calculate the residuals for the data points (0,−5)(0,-5)(0,−5), (1,−1)(1,-1)(1,−1), (2,15)(2,15)(2,15) and comment on whether the model fits the data exactly.
A researcher proposes y=ax3+bx2+cx+5y = ax^3 + bx^2 + cx +5y=ax3+bx2+cx+5 to fit (1,8)(1,8)(1,8), (2,21)(2,21)(2,21), (3,50)(3,50)(3,50), (4,101)(4,101)(4,101). Find a,b,ca,b,ca,b,c and assess the fit.
Given four observations (x,y)(x,y)(x,y): (1,2)(1,2)(1,2), (2,17)(2,17)(2,17), (3,56)(3,56)(3,56), (4,129)(4,129)(4,129), determine if a unique cubic interpolant exists and describe how to find it.
A set of five points lies approximately on y=ax3+bx2+cx+dy=ax^3+bx^2+cx+dy=ax3+bx2+cx+d. Explain how you would assess the goodness of fit without computing full regression.
Given the points (1,4)(1,4)(1,4), (2,15)(2,15)(2,15), (3,40)(3,40)(3,40), (4,85)(4,85)(4,85) and the model y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + dy=ax3+bx2+cx+d, determine whether there is an exact fit and find the coefficients if it exists.
Given the data points (1,−1)(1, -1)(1,−1), (2,5)(2, 5)(2,5), (3,19)(3, 19)(3,19) and (4,49)(4, 49)(4,49) and the model y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + dy=ax3+bx2+cx+d, determine the coefficients aaa, bbb, ccc and ddd.
Data: (1,6)(1,6)(1,6), (2,20)(2,20)(2,20), (3,54)(3,54)(3,54), (4,110)(4,110)(4,110). Test fit to y=2x3+bx2+cx+dy = 2x^3 + bx^2 + cx + dy=2x3+bx2+cx+d by finding b,c,db,c,db,c,d and checking consistency.
A data set has points (0,2)(0,2)(0,2), (1,4.5)(1,4.5)(1,4.5), (2,15.5)(2,15.5)(2,15.5), (3,40.5)(3,40.5)(3,40.5), (4,85)(4,85)(4,85) and is proposed to follow y=ax3+bx2+cx+dy = ax^3 + bx^2 + cx + dy=ax3+bx2+cx+d. Use least squares (normal equations) to set up the system for a,b,c,da,b,c,da,b,c,d.
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Question Type 1: Determining best fit model for a given set of data
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Question Type 3: Comparing two different models over the same data using SSR