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
Lesson Plan Practice Test Practice Test Answers

Chapter 3: Regression Analysis

AP Statistics Standards

I. Exploring Data:  (continued)

D. Exploring bivariate data

  1. Analyzing patterns in scatterplots
  2. Correlation and linearity
  3. Least-squares regression line
  4. Residual plots, outliers, and influential points

Objectives

Essential Question: How can we establish and quantify a cause and effect relationship between two variables?

Chap 3     2 Variable (Bivariate) Relationships

  1. Identify the response and explanatory variables from a plot.
  • response: y-variable, dependent
  • explanatory: x-variable, independent
  1. Identify positive and negative associations from scatter plots.

Note: an association does not establish cause and effect

  1. Detect linear and non-linear relationships using scatter plots.

Note: ALWAYS make a scatter plot when analyzing bivariate data

  1. Judge the relative strength of a relationship by the amount of scatter around the curve of best fit.
  2. Identify outliers on scatter plots.
    • Within the expected range X-values
    • Outside the expected range of Y-values for a given X-value
  1. Identify "influential outliers" on scatter plots.
    • Outside the expected range X-values
    • Note: in this region the expected range of Y-values is undefined
  1. Make scatter plots using the TI-83 calculator and Excel.

  2. State why any analysis of 2 variable (bivariate) data should always begin with a scatter plot regardless of which tools are used to further analyze the data.

  • identifies outliers

  • reveals gaps and clusters in the data

  • displays patterns such as linearity or non-linearity

Note: in the ideal situation all the data points would have equal influence and be uniformly distributed.

Homefun (formative/summative assessment): Exercises 1, 3, 5, 7 pp. 158-159

Relevance: Many scientific constants and predictions are based on measurements of the slopes of lines.

Essential Question: Why is it important to quantify correlation instead of just estimating it by looking at a graph?

Correlation

  1. State the meaning of correlation and how it is typically indicated.
    • r = correlation coefficient, range goes from -1 to +1
    • Strength -- the absolute value of r is close to 1
    • Direction
    • Assumes a linear relationship
  1. Calculate r using the formula:
r =     1     

Σ

(  xi - xbar  ) (  yi - ybar )  
  n - 1 sx  sy  

Be as one with the following facts about correlation:

  • r-square is bullet-proof
    • adding a constant to either y-variable or x-variable or both has no effect on r-square or slope.
    • multiplying either the y-variable or the x-variable or both has no effect on r-square
  • r is dimentionless in other words it has no units.

  • Correlation makes no distinction between explanitory and response variables.

Homefun (formative/summative assessment): Exercise 9, 15, 17 pp.159-160

Essential Question: Why would we need to find a mathematical relationship between variables? Isn't correlation enough?

Regression

  1. Explain the difference between correlation and regression.
  2. correlation: denotes the strength of an association

    regression: yeilds a mathematical model (regression equation) of the association.

  3. Perform regression/correlation analysis with the TI-83 calculator and Excel Spreads sheets.

  4. What type of error does least squares regression minimize?
  • Error measured in y-dimension (y = response variable)

  • x-dimension (explanatory variable) considered error-free

  1. Interpret regression equations.
    • Single     yhat = ax + b
    • Multiple  yhat = ao + a1x1 + a2x2 + ... + anxn
  1. Calculate ybar using a regression equation, given xbar.

  2. Properly state the meaning of slope according to the official statistics definition. (p155)

For every increase of one in the x-variable, the predicted y increases by the slope

  1. Properly interpret the intercept.
example: (sales) = 50 (advertising dollars) + 87
What are the sales with no advertising? Answer: the intercept or 87
  1. Describe the region where a given regression equation will give a meaningful association. within the range of x-values

  2. Define and decry the use of extrapolation. Extrapolation is the act of drawing a conclusion based on the regression line in a region significantly outside of the range of x-values. These conclusions can be highly misleading.

example: (bushels tomatoes) = 2 (lb fertilizer) + 10 ,
x-range 0 to 5
If Bob puts 100 lb of fertilizer on his plants, how many bushels of tomatoes will he get. Answer: zero--he kills his tomato plants.
  1. Be aware that the point (xbar, ybar) is in the center of the regression line. ybar = b (xbar) + a

Homefun (formative/summative assessment): Exercise 35, 37, 41 p.191

Essential Question: What happens to the regression analysis when we change units?
  1. Solve problems using the following equations (b = slope, a = intercept):

       b =  r (sy/ sx)                 a = ybar - b(xbar)

Action

Effect
(Derived from the above 2 equations & info at right)
Effect
(Based on review information from previous chapters.)
Slope Intercept sy sx ybar xbar

Multiply by constant = k

           
x-data points multiply by 1 / k none none multiply by k none multiply by k
y-data points multiply by k multiply by k multiply by k none multiply by k none
both none multiply by k multiply by k multiply by k multiply by k multiply by k

Add a constant = k

           
x-data points none adds -bk none none none adds k
y-data points none adds k none none adds k none
both none adds (k-bk) none none adds k adds k

 

Homefun (formative/summative assessment): Exercise 47 p.192

Relevance: Regression and correlation are the mathematical tools much of the social sciences as well as business tools are founded on.

 

 
Essential Question: What does R-Square really mean?

The Meaning of R-Square

  1. State the meaning of SST and SSE. Use them to calculate R-square.
  • SST = ∑ (yi - ybar)2    SST (Sum of Squares Total) is a measure of the scatter or variability of the y-data points about the y-data's mean.

  • SSE = ∑ (yi - yhat)2    SSE (Sum of Squared Errors) is a measure of the scatter or variability of the y-data points about the regression line. Remember, the x-data is assumed to be error free.

  • (SST - SSE) is a measure of the amount of variability in the y-data points explained by the regression line.

  • r2 = (SST - SSE) / SST is a measure of the fraction of the variation in the values of y that is accounted for by the regression line of y on x.

  1. Give the official interpretation of r-square (coefficient of determination).
    • Use the proper magic words p.180: r2 is the fraction of the variation in the values of y that is accounted for by the regression line of y on x.

    • r-square evaluates the entire equation

  1. Explain why care must be taken in using the official interpretation of r-squared. Remember, even when correlating data from random sources, r-squared can sometimes be reassuringly high.

    • Susceptible to outliers, especially influential outliers

    • Data points furthest from the center of the line have more influence. It's similar to a playground see-saw or teeter-totter: a person seated on the end will have more influence than a person seated close to the middle.

    • There may be no causative relationship between explanatory and response variables. A high r-square does not establish causation!

    • r-square applies to linear relationships. A low r-square value does not establish that there is no association. The association could be non-linear.

 

Homefun (formative/summative assessment): Homefun (formative/summative assessment): Exercise 53, 59, 63 p.192-194

Relevance: Sometimes major political decisions are made or social theories proposed based on questionable evidence from correlation/regression analysis. It's difficult to evaluate this evidence without knowing something about the meaning of r-square.

 

 

Stats Investigation (formative/summative assessment): Meaning of R-Square - time approx 2 class periods (individual work)

Purpose: Determine if a regression analysis using random numbers can yield an r-square value of 50% or more.

Instructions: Set up a regression analysis in Excel using integer x-values from 0 to 9. Use a random number from 0 to 10 for the y-values. Run this simulation 100 times. Calculate the average r-square and record the highest r-squared value. Record the three highest r-square values obtained in the class.

Save the data sets from your 4 regression/correlation  results with the highest R-square value. You will use it again at the end of the year.

Questions /Conclusions:

  1. Based on your data, does a high r-square value by itself indicate a meaningful association or causation?
  2. Is the random number generator used in this investigation truly random?
  3. Is it possible to get a high r-squared value merely from random events?
  4. What does it really mean when we say that r-square represents the fraction of the variation in the values of y that is "explained" by the least squares regression of y on x? Discuss things like the SSM and SSE.

 

Essential Question: Can a regression equation with a high R-square be inappropriate?

Residuals

  1. Define what is meant by a residual.

Mathematically: resid = yi - yhat

English: a residual is the difference between the measured y-value and the y-value predicted by the regression equation.

  1. Calculate residuals using a TI-83 calculator.

  2. State 2 ways to plot residuals.

  • Residuals vs x  commonly used with straight line equation

  • Residuals vs y  commonly used with multiple regression analysis

  1. State the major assumptions concerning distribution of y-data points about regression lines.

    Y-data normally distributed: If y-data points were repeatedly gathered for a given x-value, the y-data would form a normal distribution with its mean corresponding to the yhat value calculated with the given x-value. Remember, y-values have random measurement errors in them. Repeated measurements of a y-value will not give the exact same number.

    Uniform spread in y-data from one end of the line to the other: The spread in the above distribution would be the same for every possible x-value. (See objective 32 to estimate the size of the spread.)

  1. Interpret residual plot patterns.

Residual Plot conclusion: either appropriate or inappropriate

    • Random--appropriate
    • Smiley or Frowning Face (Mr. R's Terms)--inappropriate
    • Pattern in the scatter--inappropriate

Note: residual plots merely magnify the patterns that can be observed in a scatter plot. The horizontal line at the origin of a residual plot represents the regression line. A person skilled at interpreting scatter plots will arrive at the same conclusions that can be drawn from a residual plot.

  1. Make residual plots using a TI-83.
  1. Store x-data in L1 and y-data in L2
  2. First perform the regression analysis for L1, L2
  3. 2nd

    STAT

     RESID 

    ENTER

     RESID 

    STO

     L3
  4. Create a scatter plot of L1 on the horizontal axis and L3 on the vertical axis
  1.  State the sum of the residuals. zero

  2. Correctly interpret the standard error of the least squares regression line. The standard error of the least squares regression line is related to the residuals as shown below and is a measure of the spread of the data around the regression line. It can be considered an estimate of the standard deviation of the normal distribution described in objective 28.

    Most computer printouts will report a value for s. (see Minitab Output )

s = [ S(y - yhat)2 / (n-2) ]1/2

s = [ S(residual)2 / (n-2) ]1/2

s = [ SSE / (n-2) ]1/2

 

Homefun (formative/summative assessment): prob. 46, 60, 61, 71, 73 pp.192-196

Relevance: Even though the world is largely non-linear, parts of it can often be accurately described with linear models. Knowing when a linear model is inappropriate is essential to building effective models.

Various types of regression models are used in everything from predicting grades on AP tests to computer control of chemical plants.

 

Essential Question: How can I make an "A" on the test?

Regression/Correlation Analysis Review

  1. Work the practice test.
  2. Review the objectives.
  3. Look over free response problems from previous years.
  4. Master the vocabulary (see example below).

 

Summative Assessment: Test--Objectives 1-32

 

Stats Investigation (formative/summative assessment): Determining if a Regression Equation is Appropriate - time approx 1 class periods (individual work)

Purpose: Determine if a linear regression equation is appropriate for two different situations.

Background: Commercial resistors follow ohm's law while light bulbs, due to their high temperatures do not. Ohm's law is as follows:

I = (1/R) V

Where: I = current, V= voltage and R = resistance.

Plotting I vs. V will theoretically yield a straight line passing through the origin.

 

Instructions: Set up a least squares linear regression analysis in Excel to find the association between current (response variable) and voltage (explanatory variable) for a commercial resistor and for a light bulb. Remember that this means a scatter plot as well as finding the slope, intercept, and R-square for the data. Set up the formulas needed to plot a residual plot and make such a plot for the two sets of data.

Questions /Conclusions:

  1. Based on your data, does a high r-square value by itself indicate a meaningful association or causation?
  2. Find the resistance value in Ohms for the commercial resistor?
  3. Is a linear equation appropriate for the commercial resistor? How about the light bulb. Explain your answers.

The Practice of Statistics, Yates, Moore, McCabe

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