Mr. Rogers - AP Statistics Objectives

Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 1 Distributions 2 N-Distribution 3 Regression 4 NL Regression 5 Data

Unit Plan Practice Test Practice Test Answers

Chapter 4: Nonlinear Regression AP Statistics Standards I. Exploring Data: Observing patterns and departures from patterns (continued) D. Exploring bivariate data Transformations to achieve linearity: logarithmic and power transformations
Objectives Essential Question: Is everything we'd like to study and model linear? Chapter 4 : 2 Variable Data Continued Modeling Exponential Data Explain how data can be transformed so that linear regression produces an exponential function. First: convert all y data points to ln y. On a TI-83 calculator, if y-data is stored in L1 ans x-data is stored in L1, do LN L2 sto L4 Second: do linear regression for L1, L4 Finally: manipulate the data as shown below. ln y = ax + b e (ln y) = e[ax + b] y = [ eax ] [ eb ] y = [ eb ] [ eax ] let eb = K y = keax   Give examples where an exponential regression model would be appropriate. Growth or decay situations (response variable multiplied by a fixed amount in each time interval) such as: bacteria population growth the doubling of computer power (Moore's Law) Growth in wind power (wind power map) radioactive decay (decay of a population of atoms) Explain how to determine if an exponential model is appropriate. theoretical basis such as objective 2 above random residuals (this means exponential regression is appropriate. It does not necessarily mean it's right.) Explain why an exponential model should not be selected on the basis of optimizing r-square. A different form of  non-linear equation may have a higher R2 value but be less appropriate. Perform exponential regression on a TI-83 calculator using the NON-transformed data and note that these results and the results obtained with transformed data are mathematically the same. NON-transformed data: y = abx Transformed data:  y = keax y = k(ea)x Let: k = a ,  ea = b By substitution: y = abx Homefun (formative/summative assessment): Read section 4.1; prob. 4.7, 4.11 Relevance: Exponential data is commonplace in many business, biological, chemistry, physics, and other  areas. Knowing how to deal with it and how to model it is a significant career skill.	Activities Lesson 1 Key Concept: Transforms Purpose: Create a linear plot from nonlinear data Seat Work: have students graph an exponential example and do linear regression and residuals. Interactive Discussion: Objectives. Explain the terms concaved upward and downward. Review exponents logarithms and explain. Seat Work: perform the transform on the above data and derive an exponential model from it. Compare this with the linear model. Time Microbes 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256
Essential Question: If a mouse weighing 0.5 lb were scaled up by a factor of 100, how much would it weigh? Modeling Power Function Data Explain how data can be transformed so that linear regression produces a power function. First: convert all x and y data points to ln x and ln y. On a TI-83 calculator, if y-data is stored in L1 and x-data is stored in L1, do LN L1 sto L3 and  LN L2 sto L4 Second: do linear regression for L3, L4 Finally: manipulate the data as shown below. ln y = a(lnx) + b e (ln y) = e[a(lnx) + b] y = [ e(lnx)a ] [ eb ] y = [ eb ] [ xa ] let eb = K y = kxa   Give examples where a power regression model would be appropriate. Definition of scaling factor: If an object is to be scaled up to a larger size without changing the appearance of the object, all the dimensions of the object have to be multiplied by a common factor. This factor is called the scaling factor. Scaling problems: Volume & mass scale with the cube of the scaling factor Area scales with the square of the scaling factor Explain how to determine if a power model is appropriate. theoretical basis such as objective 6 above random residuals Explain why a power model should not be selected on the basis of optimizing r-square. A different form of  non-linear equation may have a higher R2 value but be less appropriate. Perform power regression on a TI-83 calculator using the NON-transformed data and note that these results and the results obtained with transformed data are the same.   Homefun (formative/summative assessment): prob. 4.13, 4.15 Relevance: Power-functiondata is commonplace in many business, biological, chemistry, physics, and other  areas. Knowing how to deal with it and how to model it is a significant career skill.	Activities Lesson 2 Key Concept: The transform needed for a power model Purpose: Recognize the situations where a power model is appropriate and create one. Interactive Discussion: Objectives. Define scaling factor. Demonstrate how scaling factors work using spheres, cubes and rectangular prisms. Seat Work: plot following simulated data Pumpkin Dia. Surface Area 1 1.2 2 4.7 3 9.0 4 16.9 5 25.1 6 36.5 7 49.0 8 64.9 Perform linear regression analysis and find R2. Transform both x and y data. repeat the process. Convert the linear regression equation  to a power model Work Example 4.9 Fishing Tournament p.216   For more information about scaling and why it's incredibly important Read: Insultingly Stupid Movie Physics Chapter 4, Scaling Problems: Big Bugs and Little People, pp 51 - 66
Essential Question: What is the most common form of extrapolation? Interpreting Correlation and Regression Decry the evils of extrapolation but also be aware that it's commonly used. projected sales--in order to plan ahead, companies will often attempt to predict the next year's sales and earnings based on regression analysis of data from previous years. projected population growth projected impact of advertising dollars spent--used for determining what the future advertising budget should be. radioactive dating--there's sound theory backing radioactive dating but obviously no one collected data on it thousands of years ago. Be aware of ways the risks of extrapolation can be moderated. Sound theoretical basis for the regression model Strong supporting data from independent sources. For example: a limited amount of extrapolation using the recent exponential grow in American wind-power electrical generation is reasonable based on ready availability of wind resources, low cost compared to other forms of generation, and concerns about global warming, 3 factors which make wind-power attractive. Extrapolating only slightly beyond the range of the actual data. The greater the distance beyond the data's range, the greater the risk. Simple regression model such as linear, exponential, or power. Extrapolation with high order polynomials is very dangerous. Positive results with various indicators such as outlier-free scatter plot, high r-square, random residuals, etc. Identify possible lurking variable. An important variable which is not included in the study. Name the most common lurking variable. time) State the pitfall of using averaged data. It makes the r-squared value higher. Hence, the results look better than they really are. Homefun (formative/summative assessment): prob. 4.27, 4.32	Lesson 3 Key Concept: Extrapolation and lurking variables   Purpose: Understand how conclusions drawn from data can be disastrously wrong Interactive Discussion: Objectives. Explain real growth curves--usually sigmoidal. Seat Work: Uncover the lurking variable of time in the example on p. 228. Plot math classes per student vs. time.   Work problem 4.19 on p.222.
Essential Question: Can we ever be completely sure that causation exists? Causation In other words, is the association between the x and y variables due to  the x-variable actually causing a response in the y-variable. State 4 possible explanations for getting a strong association based on regression/correlation analysis. Causation --Sometimes it's true: x causes y Common response variables (affect both x & y variables), example: rum (y) and Methodist Ministers (x) are both affected by the common response variable, population growth (z). Confounding variables (affect the y variable but not the x), example: The shaman chants an incantation (x) and five days later the patient who seemed near death gets well. The patient's immune system (z) was the real cause. Random chance (the association is temporary in nature and is the result of numerous unidentifiable factors that are not reproducible), example: Bob finds a 1957 penny on the sidewalk as he enters the casino. When he subsequently wins $2000 dollars at roulette, he concludes that the penny is his good luck charm. Explain 4 steps toward establishing causation. Generally all 4 steps are required especially for controversial situations. Carefully controlled experiments--the gold standard. Can sometimes be as simple as turning the causative variable on and off. Weakness = experiments often are run in an artificial environment. Multiple independent observational studies of different types Account for, control, or eliminate lurking variables--Must be done in both observational and experimental studies. Accounting for lurking variables usually means including them in multiple linear regression analysis. Develop a plausible theory--without a plausible theory, even experimental data can be questioned. Homefun (formative/summative assessment): Read section 4.2; prob. 4.35, 4.41, 4.45 Summative Assessment: Test objectives 1-14 and previous regression/correlation objectives	Lesson 4 Key Concept: Extrapolation and lurking variables   Purpose: Understand how conclusions drawn from data can be disastrously wrong Interactive Discussion: Objectives.   Questions The dog barked and the tree fell down. Did the dog cause the tree to fall? What are the possible common response variables?  What are the possible confounding variables? Could the two events coincide due to random events? Could the tree-felling dog be tested in an experiment? Is there a plausible theory for why the tree could be felled by the noise of a dog barking?   Video: use video on smoking
AP Statistics Standards I. Exploring Data: Observing patterns and departures from patterns (continued) E. Exploring categorical data 1. Frequency tables and bar charts 2. Marginal and joint frequencies for two-way tables 3. Conditional relative frequencies and association 4. Comparing distributions using bar charts
Essential Question: How can categorical data be represented and interpreted? Categorical Data Create frequency tables for categorical data. Convert the above tables into bar charts. Use conditional distributions based on relative frequencies to establish associations. Compare distributions using bar charts. Interpret 2-way tables. 2 variables Convert to % Interpret marginal distributions. 2 for each table, horizontal & vertical Histogram like Single variable only Calculate and interpret conditional distributions   Essential Question: Can data sets be added together to obtain a larger sample size and hence more meaningful conclusion?    Simpson's Paradox Analyze data for Simpson's paradox. Conclusions based on parts can be reversed when considering the whole Conclusions based on parts is more likely to be valid. State two conditions which must exist for Simpson's Paradox to occur. One or more lurking variables Data from unequal sized groups being combined into a single group. State how Simpson's paradox can be prevented. Avoid combining data from unequal groups into a single study Identify and include lurking variables in the study   Homefun (formative/summative assessment): Read Simpsons's Paradox - When Big Data Sets Go Bad prob. 4.37, 4.39, 4.45	Lesson 5 Key Concept: Simpson's Paradox   Purpose: Understand how conclusions drawn from data can be disastrously wrong Interactive Discussion: Objectives. Work through hospital example of Simpson's paradox. Individual work: Work through Simpson's paradox worksheet provided by teacher. Use Titanic data to determine if the class of one's ticket had an association with the chances of survival. Materials: Simpson's Paradox Worksheet and Titanic data. p. 247
Essential Question: Can data sets be added together to obtain a larger sample size and hence more meaningful conclusion?   Analyze data for Simpson's paradox. Conclusions based on parts can be reversed when considering the whole Conclusions based on parts is more likely to be valid. State two conditions which must exist for Simpson's Paradox to occur. One or more lurking variables Data from unequal sized groups being combined into a single group. State how Simpson's paradox can be prevented. Avoid combining data from unequal groups into a single study Identify and include lurking variables in the study Homefun (formative/summative assessment): Read Simpsons's Paradox - When Big Data Sets Go Bad prob. 4.37, 4.39, 4.45   Summative Assessment: Test Objectives 1 - 27	Lesson 5 Key Concept: Simpson's Paradox   Purpose: Understand how conclusions drawn from data can be disastrously wrong Interactive Discussion: Objectives. Work through hospital example of Simpson's paradox. Individual work: Work through Simpson's paradox worksheet provided by teacher. Use Titanic data to determine if the class of one's ticket had an association with the chances of survival. Materials: Simpson's Paradox Worksheet and Titanic data.