Lesson 19: Cautions about Correlation and Regression

Review:

• Pearson Correlation Coefficient
• R-Squared
• Exam 3 on Wed, Nov 20
• Review for Exam 3 on Mon, Nov 18

Presentation:

• Cautions about Correlation and Regression
  • Influential Observations
    • Outliers
    • Diabetes and blood sugar (p. 129-130)
  • Lurking Variables
    • Explanatory variables not included
    • High School math and success in college (p. 132-133)
  • The Question of Causation
    • Correlation does not imply causation
    • Spurious Correlations
    • Retrospective study = looking back to find possible causes for an established outcome among a sample population
    • Prospective studies = following a sample population over time and studying behaviors possibly linked to likelihood of an outcome
    • Video
  • Extrapolation
    • Use of regression for prediction far outside the range of the explanatory variable
    • Predictions outside the range are unreliable
    • Further outside the range = less reliable predictions
  • Sensitivity Analysis Demonstration
    • Remove influential observations and recalculate regression equation and R-Squared
    • Use Pueblo Voter Turnout data

• Produce a scatter plot with Time Period on the x axis and Ballots Cast on the y axis
• Find the linear regression equation and the corresponding R-Squared value
• Remove the 2004 data
• Recalculate the linear regression equation and the corresponding R-Squared value
• Remove the 2016 data
• Recalculate the linear regression equation and the corresponding R-Squared value

Activity:

• Produce a scatter plot with Price of Coffee on the x axis and Deforestation on the y axis
• Find the linear regression equation and the corresponding R-Squared value
• Estimate Deforestation % assuming Price of Coffee is $0.90 per pound.
• Sensitivity Analysis
  • Identify and remove the most influential observation (use scatterplot)
  • Recalculate the linear regression equation and the corresponding R-Squared value
  • Recalculate Deforestation % assuming Price of Coffee is $0.90 per pound.
    

Lesson 18: Pearson Correlation Coefficient and R-Squared

Review:

• Quiz 6
• Linear Regression with Professor Kuan
• Exam 3 on Wed, Nov 20
• 2020 Pueblo Voter Turnout Estimates

Presentation:

• Correlation and R-Squared
  • Pearson Correlation Coefficient
    • r = SSxy/√(SSxx*SSyy)
    • -1 < r < 1
  • R-Squared =(r)^2
  • Example
    • Use Beer Party
    • Sample data: {(2,3), (4,6), (7,9), (11,12)}
    • Calculate Pearson Correlation Coefficient (r)
    • Calculate R-Squared

Activity:

• Use the data above
  • Make a scatterplot
  • Calculate the Sum of Squares
  • Calculate the linear regression equation
  • Calculate the Pearson Correlation Coefficient
  • Calculate R-Squared

• Use the data above
  • Make a scatterplot
  • Calculate the Sum of Squares
  • Calculate the linear regression equation
  • Calculate the Pearson Correlation Coefficient
  • Calculate R-Squared

Lesson 17: Linear Regression

Review:

• Scatterplots
• Sum of Squares

Presentation:

• Video
• Linear regression (simple, bivariate)
  • Calculate equation of the regression line
    • y-hat = b1*x + bo
      • b1 = “slope” of the line
      • b0 = “y-intercept”
    • Calculate Sum of Squares: SSxx, SSyy, SSxy
    • b1 = SSxy/SSxx
    • b0 = (∑y/n) – b1*(∑x/n)
  • Example
    • Beer party data: {(60,10), (70,12), (80,20), (90,40)}
    • Calculate slope (b1) and y-intercept (b0) for linear equation
    • Estimate Beer consumption when temperature is 75 degrees
    • Download Complete Example

Activity: Submit this work before leaving class!!

• Forecast 2020 voter turnout in Pueblo County using the data below
  • Create a Scatterplot
  • Calculate Sum of Squares: SSxx, SSyy, SSxy
  • Calculate slope and intercept for the regression line and write the linear equation
  • Use the regression equation to forecast Ballots Cast in 2020
    • Hints
      • Use Time Period as the independent (x) variable
      • Use Ballots Cast as the dependent (y) variable
      • To forecast 2020 turnout, solve for y-hat when x = 5

Assignment:

• Study for Quiz on Monday covering Scatterplots, Sum of Squares and Linear Regression

Lesson 16: Least Squares Regression

Review:

• Scatterplots

Presentation:

• Least Squares Regression
  • Find a line of best fit for a set of points
  • Least Squares Regression Proof: MethodLeastSquares
  • Example: Beer Party
    • Use temperature to predict 12-packs consumed
    • Data from previous parties (Temp degrees F, 12-packs consumed)
      • Party 1: 60°, 10 12-packs
      • Party 2: 70°, 12 12-packs
      • Party 3: 80°, 20 12-packs
      • Party 4: 90°, 40 12-packs
    • Scatter Plot
    • Table setup
  • Sum of Squares calculations
    • SSxx = ∑x² – (∑x)²/n
    • SSyy = ∑y² – (∑y)²/n
    • SSxy = ∑xy – (∑x*∑y)/n
    • Demonstrate Calculations
      • Beer party data: {(60,10), (70,12), (80,20), (90,40)}
      • Another example: {(2,3), (4,6), (7,9), (11,12)}
  • We will use these Sum of Squares calculations to determine
    • linear equation
    • Pearson correlation coefficient
    • R-squared

Activity:

• A study was conducted on mercury (Hg) concentrations in fish taken from Lake Natoma in California. The researchers were concerned that mercury concentration levels in sample fish tissue might differ depending on the lab testing the fish. Fish tissue samples from 10 largemouth bass were sent to two labs, Columbia Environmental Research Center (CERC) and University of California, Davis (UC Davis), for inter-laboratory comparison. Mercury concentration is measured in micrograms of mercury per gram of fish tissue (dry weight). Use the mercury concentration data below to Calculate Sum of Squares (SSxx, SSyy, SSxy).

• Satellites are one of the many tools used for predicting flash floods, heavy rainfall, and large amounts of snow. Geostationary Operational Environmental Satellites (GOES) collect data on cloud top brightness temperatures (measured in degrees Kelvin (°K)). It turns out that colder cloud temperatures are associated with higher and thicker clouds, which could be associated with heavier precipitation. Data consisting of cloud top temperature measured by a GOES satellite and rainfall rate measured by ground radar appear in the table below. Because ground radar can be limited by location and obstructions, having an alternative for predicting the rainfall rates can be useful. Use the temperature and radar rain rate data below to Calculate Sum of Squares (SSxx, SSyy, SSxy).

Problems above from the Against All Odds video series guide for Unit 10.

Lesson 15: Scatterplots

Review:

• Exam 2 Results
• Competition results
• Register for classes
  • BUSAD 360: Advanced Statistics
  • ECON 325: Geography of World Economy

Presentation:

• Scatterplots
  • Visualize relationship between 2 variables
    • independent (x) and dependent (y)
    • explanatory (x) and response (y)
    • horizontal axis (x) and vertical axis (y)
  • Visual Correlation
    • Direction: Positive or Negative
    • Strength: Strong, Moderate, Weak
  • Examples
    • Ice Cream Sales
    • Colorado TCAP Scores and % Free-Reduced Lunch
    • Gender Wage Gap
  • Alternative Uses
    • Predicted vs Actual
    • Hot/Crazy
  • Video

Activity:

Problem 1. A study was conducted on mercury (Hg) concentrations in fish taken from Lake Natoma in California. The researchers were concerned that mercury concentration levels in sample fish tissue might differ depending on the lab testing the fish. Fish tissue samples from 10 largemouth bass were sent to two labs, Columbia Environmental Research Center (CERC) and University of California, Davis (UC Davis), for inter-laboratory comparison. Mercury concentration is measured in micrograms of mercury per gram of fish tissue (dry weight). Use the mercury concentration data below to create a Scatterplot and characterize the nature of any correlation.

Problem 2. Satellites are one of the many tools used for predicting flash floods, heavy rainfall, and large amounts of snow. Geostationary Operational Environmental Satellites (GOES) collect data on cloud top brightness temperatures (measured in degrees Kelvin (°K)). It turns out that colder cloud temperatures are associated with higher and thicker clouds, which could be associated with heavier precipitation. Data consisting of cloud top temperature measured by a GOES satellite and rainfall rate measured by ground radar appear in the table below. Because ground radar can be limited by location and obstructions, having an alternative for predicting the rainfall rates can be useful. Use the temperature and radar rain rate data below to create a Scatterplot and characterize the nature of any correlation.

Problems copied from the Against All Odds video series guide for Unit 10.