AP Statistics Chapter 4 Quiz Review

Transcription

1 AP Statistics Chapter Quiz Review NAME The data below list the life epectancy for white males in the United States for every decade during the last century (1 = 191 to 191, = , etc.). Create a model to predict future increases in life epectancy. (National Vital Statistics Report) Decade Life ep PART A - Draw a scatterplot of the above data. Use appropriate scales and labels. 7 Scatterplot of LIFE_EXP vs 7 LIFE_EXP 1 PART B Determine an appropriate model for the original data. Eponential Model: 1.9 Scatterplot of log(life_exp) vs.3 s of log(life_exp) and (response is log(life_exp)) 1.. log(life_exp) R =.91 and R =.9 Power Model: 1.9 Scatterplot of log(life_exp) vs log().1 s Versus log() (response is log(life_exp)) 1.. log(life_exp) log() log(). 1. R =.99 and R =.993

2 PART C Analysis of chosen model (1) Which of the above models should we use to represent the original data, and why? - The plot of log(y) vs. log() is more linear than the plot of log(y) vs. - The residual plot of log(y) vs. log() shows random scatter, while the residual plot for log(y) vs. shows curvature. - Finally, R =.99 for the power model while R =.91 for the eponential model. Therefore, the power model appears to be a better fit. () For the model you chose in problem (1), write the linear equation that models the relationship between the transformed data. Use correct notation and define all variables. logɵy = log, where y = life epectancy and = decade (where 1=19- ( ) ( ) 191, = , etc.) (3) Perform the appropriate inverse transformation (i.e., do the algebra) to obtain an equation for a b model representing the original data (must be of the form y = a b or y = a ). Use correct notation. logɵy = log ( ) ( ) ɵ log( ) y = 1 ɵ 1.7 log y = 1 ( 1 ) ɵy =.33 () What does the model predict as life epectancy of white males for the decade from 1-1?.197 The decade 1-1 corresponds to = 11. Then, ɵ y =.33( 11) years. () Mortality tables about the US population give a life epectancy of 7.7 years for the 1-1 decade. What is the residual for this measurement? Show the setup that leads to this answer. = actual predicted = =.3 years. Now do all of the above for the following data regarding recent baseball salaries: Player Year Salary (in millions of $) Nolan Ryan George Foster 19. Kirby Puckett Jose Canseco Roger Clemens Ken Griffey, Jr Albert Belle Pedro Martinez Mike Piazza Mo Vaughn Kevin Brown Carlos Delgado Ale Rodriguez 1. Manny Ramirez. Ale Rodriguez..197

3 PART A - Draw a scatterplot of the above data. Use appropriate scales and labels. In this case, I d transform the years to be, or {1, 3, 11, }. This will give us an easier model to work with later on. Here s the graph, using these transformed values. The plot has a nonlinear shape. Scatterplot of Salary (in millions) vs Salary (in millions) PART B Determine an appropriate model for the original data. Eponential Model: (log(y) vs ) 1. Scatterplot of log(salary) vs s Versus (response is log(salary)) 1..1 log(salary) Power Model: (log(y) vs. log()) R =.9 and R = Scatterplot of log(salary) vs log(years). s Versus log(years) (response is log(salary)) 1..1 log(salary) log(years) log(years) R =.9 and R =.

4 PART C Analysis of chosen model () Which of the above models should we use to represent the original data, and why? - The plot of log(y) vs. is more linear than the plot of log(y) vs. log() - The residual plot of log(y) vs. shows random scatter, while the residual plot for log(y) vs. log() shows curvature. - Finally, R =.9 for the power model while R =.97 for the eponential model. Therefore, the eponential model appears to be a better fit. (7) For the model you chose in problem (1), write the linear equation that models the relationship between the transformed data. Use correct notation and define all variables. logɵy =.1 +.3, where y = salary in millions and = years past 1979 ( ) () Perform the appropriate inverse transformation (i.e., do the algebra) to obtain an equation for a b model representing the original data (must be of the form y = a b or y = a ). Use correct notation. logɵy = ANS (see below 1 ): (9) What does the model predict as a salary for 1? ( ) ɵy = ɵ.3.1 y = 1 ( 1 ) ɵ y = 1.( 1.13) The year 1 corresponds to = = 31. Then, ɵ y = 1.( 1.13) million. (1) As far as I know, and in terms of salary alone, Ale Rodriguez is (and has been for the past 1 years) the highest paid player in baseball with a 1 salary of $33 million. What is the residual for his particular salary, according to our model? = actual predicted = = million dollars. 1 If you did NOT transform the years to Years beyond 1979, your model would be ɵ or upon algebraic transformation, ɵ y 1 ( 1 ) ( y) log = , =. This presents problems because our calculators will give =, which is not helpful at all. You can sort of get around this by setting Y = 1^Y1. Then your calculator evaluates the linear epression first and then uses that as an eponent to base 1. I would try to avoid doing things this way, though.

5 QUESTIONS to ask yourself: - Do you know how to create the models for both eponential and power? - Do you know how to create the residual plots in each case? - Do you know how to labels the scatterplots and residual plots in each case? - Do you know how to do the algebra to get power or eponential models from the linear models? - Do you know how to use your models to make predictions? - Do you know how to calculate residuals?