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6. In a study undertaken to model the reliability of car engines by the motor industry, the time to failure of these engines was modeled by the density function 4 f(t) = 1½te-1/r (a) Show that this density function is a member of the exponential family of distributions. Determine the natural parameter and the canonical link function, and find the variance function, V(µ). (b) A GLM with an inverse link function was fitted with a linear pre- dictor n = ai +ẞxsize, where a is the parameter for the ith car manufacturer and ẞ is the parameter for the engine size, size. The ML estimators of these parameters were calculated â₁ = -0.0712, 2 = -0.0565, 3 = -0.0678 and ẞ = −0.0207, where there existed only three car manufacturers, i = 1,2,3. Using this model, find the estimated mean survival time for a car pro- duced by manufacturer 3 with a 2.5 liter engine? Express the ratio of mean survival times for cars produced by manufacturers 1 and 2 as a function of the engine size, Isize.

6. In a study undertaken to model the reliability of car engines by the motor industry, the time to failure of these engines was modeled by the density function 4 f(t) = 1½te-1/r (a) Show that this density function is a member of the exponential family of distributions. Determine the natural parameter and the canonical link function, and find the variance function, V(µ). (b) A GLM with an inverse link function was fitted with a linear pre- dictor n = ai +ẞxsize, where a is the parameter for the ith car manufacturer and ẞ is the parameter for the engine size, size. The ML estimators of these parameters were calculated â₁ = -0.0712, 2 = -0.0565, 3 = -0.0678 and ẞ = −0.0207, where there existed only three car manufacturers, i = 1,2,3. Using this model, find the estimated mean survival time for a car pro- duced by manufacturer 3 with a 2.5 liter engine? Express the ratio of mean survival times for cars produced by manufacturers 1 and 2 as a function of the engine size, Isize.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.1: Measures Of Center
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6. In a study undertaken to model the reliability of car engines by the motor industry, the time to failure of these engines was modeled by the density function 4 f(t) = 1½te-1/r (a) Show that this density function is a member of the exponential family of distributions. Determine the natural parameter and the canonical link function, and find the variance function, V(µ). (b) A GLM with an inverse link function was fitted with a linear pre- dictor n = ai +ẞxsize, where a is the parameter for the ith car manufacturer and ẞ is the parameter for the engine size, size. The ML estimators of these parameters were calculated â₁ = -0.0712, 2 = -0.0565, 3 = -0.0678 and ẞ = −0.0207, where there existed only three car manufacturers, i = 1,2,3. Using this model, find the estimated mean survival time for a car pro- duced by manufacturer 3 with a 2.5 liter engine? Express the ratio of mean survival times for cars produced by manufacturers 1 and 2 as a function of the engine size, Isize.
Transcribed Image Text:6. In a study undertaken to model the reliability of car engines by the motor industry, the time to failure of these engines was modeled by the density function 4 f(t) = 1½te-1/r (a) Show that this density function is a member of the exponential family of distributions. Determine the natural parameter and the canonical link function, and find the variance function, V(µ). (b) A GLM with an inverse link function was fitted with a linear pre- dictor n = ai +ẞxsize, where a is the parameter for the ith car manufacturer and ẞ is the parameter for the engine size, size. The ML estimators of these parameters were calculated â₁ = -0.0712, 2 = -0.0565, 3 = -0.0678 and ẞ = −0.0207, where there existed only three car manufacturers, i = 1,2,3. Using this model, find the estimated mean survival time for a car pro- duced by manufacturer 3 with a 2.5 liter engine? Express the ratio of mean survival times for cars produced by manufacturers 1 and 2 as a function of the engine size, Isize.
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