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MATH431 Linear Regression

Questions:

Analyze linear regression models to determine the nature of the relationship between sets of predictor variables and a particular response variable. Apply logistic regression, Poisson regression, and general linear regression models as nonlinear regression models when response outcomes are discrete and error terms are not normally distributed. At a minimum, your summary should include: What was learned during the weeks seven and eight.

A summary of your thoughts on the material covered during the weeks. Identify a practical application (not a problem, but an application) that uses one or more of the concepts discussed this week. Show your work in your solution to the practical application Links to any web sites used to help support learning and understanding of the materials for the week. (Outside of the tutorials) Please be detailed in your practical example.

Explain which formulas or concepts used and what the results are. If you have difficulties with a good practical example, look at the applications in the text book and see if you can derive similar experiences. Below is a weekly summary from a student that got full credit. This is what is expected in your weekly summary. Do not short change yourself on this assignment. These are easy points. What I learned this week:

Per our textbook's theorem and excluding degenerate cases, the equation A x 2 + C y 2 + D x + E y + F = 0 (where A and C ? 0), by the product of AC defines a parabola if the result is zero, an ellipse if the result is greater than zero, and a hyperbola if the result is less than zero. I also learned how to use a rotation of axes to transform equations using the definitions of sine and cosine. Also, I learned that a rotation through an appropriate angle can transform any equation of the form of the (aforementioned) equation into an equation in x' and y' without an x'y' -term.

To transform the (above) equation into an equation in x' and y' without an x'y' -term, rotate the axes through an angle ? that satisfies the equation cot (2?) = (A-C)/B My thoughts on the material: After reviewing the lesson of the week, I came to the understanding that ultimately utilizing a rotation of axes to transform equations permits us to pursue correlations that will allow us to express x and y in terms of x' , y' and ?. This led to the formation of the rotation formulas theorem. Practical Application: An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. The path of a gear in a piece of mechanical equipment is given by the equation 7 x 2 + 8 xy +7 y 2 – 6 = 0 Determine the appropriate angle that will allow the other gear to turn while maintaining contact at all times. Using the theorem, we can first identify the conic section associated with this equation, by finding the result of AC . In this case, A = 7 and C = 7, so AC = 49, which, per the theorem, defines an ellipse because AC is greater than zero. Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 7 x 2 + 8 xy +7 y 2 – 6 = 0 The rotation formula theorem states that if the x - and y - axes are rotated through an angle ?, the coordinates ( x , y ) of a point P relative to the xy-plane and the coordinates ( x ', y ') of the same point relative to the new x '- and y ' axes are related by the formulas x = x' cos ? – y' sin ? y = x' sin ? – y' cos ?

To determine the angle ? that, in conjunction with the rotation formulas, will eliminate the term 8 xy from the given equation, we use the transformation theorem which states that To transform the equation A x 2 + B xy + C y 2 + D x + E y + F = 0, B ? 0 into an equation in x ' and y ' without an x ' y '-term, rotate the axes through an angle ? that satisfies the equation \f[cos(2\theta)=\frac{A-C}{B}\f] So, we will use the following rules to choose an acute angle ? If cot (2?) ? 0, then 0° < 2? ? 90°, so 0° < ? ? 45° If cot (2?) < 0, then 90° < 2? < 180°, so 45° < ? < 90°. Each of these results in a counterclockwise rotation of the axes through an angle ?. We find ? using A = 7, B = 8, and C = 7. cot (2 ?) = (7-7)/8 = 0 Since cot (2?) = 0, 2? = 90° and ? = 45°. Find cos ?. cos(45°) = \f$\frac{\sqrt{2}}{2}\f$ Find sin ?. sin(45°) = \f$\frac{\sqrt{2}}{2}\f$ Now we can determine the rotation formulas. x = x' cos ? – y' sin ? = x'\f$\frac{\sqrt{2}}{2}\f$ – y'\f$\frac{\sqrt{2}}{2}\f$ = \f$\frac{\sqrt{2}}{2}\f$ (x' – y') y = x' sin ? + y' cos ? = x' \f$\frac{\sqrt{2}}{2}\f$ + y'\f$\frac{\sqrt{2}}{2}\f$ = \f$\frac{\sqrt{2}}{2}\f$ (x' + y') This practical application supports the fact that two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times.

Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle. An interesting website that I visited (outside of those listed in our Content section and in the syllabus) is Real-Life Applications — Conic Sections. It includes a fascinating slide show of several real-life applications of conic sections. I invite you all to give it a look if you have the time.