Mr. Rogers - AP Statistics Objectives
Unit Plan Practice Test Practice Test Answers
      AP Statistics Super Splendid Non Linear Regression Practice Test 
  1 ) A common response variable influences __________________. both the x and y or response and explanatory variables
  2 ) Residuals can be mathematically defined as follows: resid = yi - y-hat
  3 ) When a power regression is performed, which variable(s) is/are transformed? Both are transformed. The y or response variable is transformed into Ln (y) and the x or explanatory variable is transformed into Ln (x).
  4 ) Name 4 ways to establish causation   
  • Carefully controlled experiments
  • Multiple independent observational studies of different types
  • Account for, control, or eliminate lurking variables
  • Develop a plausible theory
  5 ) Bob looks at a regression analysis with an r-square of 0.11, a slope of .17, and an intercept of 6, and concludes that there is definitely no relationship between the variables. Is this a proper conclusion? Explain. No, there could be a non-linear association
  6 ) Write the official Statistics definition of slope. For every increase of one in the x-variable, the predicted y increases by the slope
  7 ) For a least squares regression, If Sx = 8, r^2 = 0.64, and the slope = 100, what does Sy equal?  Sy = 1000
  8 ) Which is smaller SST or SSE? Usually SSE but they could be equal if R-square = 0
  9 ) Name a situation where an exponential regression would likely be appropriate. Growth/decay.
  10 ) When an exponential regression is performed, which variable(s) is/are transformed? The y or response variable is transformed into Ln (y)
  11 ) Why is a high level of association based on a single regression analysis not be considered proof of causation? Lurking variables or random events may be responsible for the association.
  12 ) For a regression/correlation analysis R-square = 0.75. Using the definition of R-square, explain what the number means. The regression equation explains 75% of the variability in the y-data.
  13 ) What is the first thing that should be done when performing regression analysis. Scatter plot
  14 ) If a residual plot has a pattern in it what conclusion should you draw about the regression equation? It's inappropriate
  15 )
Convert the following into an exponential form: ln(y-hat) = 5x + 7.    
y-hat  = 1097 [ exp ( 5x ) ]
  16 )
Convert the following into an exponential form: ln(y-hat) = - 2x + 4.
y-hat  = 54.6 [ exp (-2x ) ]
  17 )
Convert the following into a power equation form: ln(y-hat) = - 4(lnx) + 2.75
y-hat  = 15.6 x^(-4)
  18 )
Convert the following into a power equation form: ln(y-hat) = 3(lnx) + 5
y-hat  = 148 x^(3)
  19  
For a least squares regression, If Sx = 12, r^2 = 0.8 and Sy = 4 equal what is the slope equal to? r = 0.9, Slope = 0.9 (4/12) or 0.3
  20 )
For a least squares regression, y-hat = 4x + 20, r^2 = 0.64,  Sy = 5. What is
r = 0.8, Sx = 0.8 (5) / 4 or 1
  21 ) For a least squares regression, y-hat = 4x + 20, r^2 = 0.64, Sy = 5, x-bar = 20. Find y-bar.   y-bar = 100
  22 ) For a least squares regression, y-hat = 4x + 20, r^2 = 0.64, Sy = 5, x-bar = 20. x is increased by 10. Find the corresponding increase in y. 40
  23 ) For a least squares regression, y-hat = 4x + 20, r^2 = 0.64, Sy = 5, x-bar = 20. Find the value of y when x = 7.   48
  24 ) State the pitfall of using averaged data r-square becomes closer to 1.0, the association appears stronger than it rally is.
  25 ) What is a confounding variable? affects only y-data
  26 ) What is a common response variable? affects x-data and y-data
 

27

) Bob decides to sell bacteria burgers for a living. (They're full of protein and easily grown.)  He has heard of linear regression and wants a linear model. Perform linear regression for him and report the results. Explain why this is not a good model. Next perform an appropriate form of non-linear regression and explain this model to him. Make sketches of the scatter diagram for the data and all residual plots. Do not forget to report and explain the r-square values.
      time:                        1 2 3 4 5 6 7 8  
        number of bacteria: 2 5 9 19 30 68 130 252  
                         
     

linear equation analysis

scatter plot: (draw a sketch of plot)
 
linear equation:  y-hat = 30.5 x - 72.9 ; r-square = 0.734
 
residual plot: (draw a sketch of plot)
 
conclusion: The scatter plot of the data indicates no outliers. The regression equation explains 73.4 % of the variability in the number of bacteria ( a reasonable level ). However, the residual plot indicates the equation is not appropriate. The equation is also not theoretically plausible since this is a growth over time situation and these usually follow an exponential relationship. Based on these findings, the linear equation should not be used to model bacteria growth vs. time.

 

     
exponential equation analysis
 
scatter plot: (draw a sketch of plot)
 
exponential equation:  y-hat = 1.15830 (1.9633)^x   or
y-hat = 1.15830 [ exp( 0.674695 x )] ; r-square = 0.997 ;
 
Residual plot: (draw a sketch of plot)
 
conclusion: The scatter plot of the data indicates no outliers. The regression equation explains 99.7% of the variability in the number of bacteria ( an improvement over the linear model ). The residual plot indicates the equation is appropriate. The equation is theoretically plausible since this is a growth over time situation. Based on all of these findings, the exponential equation is a valid model for bacteria growth vs. time.
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