Final Exam Review

Final Exam @ 10:30a on Tue Dec 10 in HSB 110

• Format 8-12 word problems, some with multiple steps/parts
• Students are permitted a 1-page “cheat sheet” for formulas and notes (1 side only)
• Bring a Z-table, t-table and a calculator

Review:

• Grades – please check your scores in Bb! Or with me.
• Complete a course evaluation
• Spring enrollment
• Complete Makeup Exam Solutions
• Review Problem Types

Problem Types:

• Describe a small data set
  • Stemplot
  • Five-number summary with boxplot
  • Mean and Standard Deviation
• Normal Curve
  • Calculate Z-scores
  • Find percentages under the normal curve (below, above, between)
  • Z table
  • Empirical Rule
  • Reverse lookup, e.g., what value of x is greater than 90% of a normal distribution?
• Estimation with Confidence Intervals
  • Point estimate
  • Margin of error (MoE) calculation
  • Point estimate +/- MoE = Confidence intervals
  • Large sample (Z table), n>30
  • Small sample (t table), n<=30
  • Proportion (p-hat) – unique standard error calculation
• Significance Testing
  • 4 steps
  • Setup Ho and Ha
  • Calculate Z statistic
  • Calculate p-value
  • Interpret results (Reject or Fail to Reject)
• Bivariate Analysis
  • Scatterplot and correlation
  • Sum of Squares
  • Linear regression
    • Slope
    • y-intercept
    • linear equation
  • Pearson correlation coefficient (r) and R-squared
  • Estimation/forecasting with linear regression
    • Use the linear equation to solve for x
    • Time series (don’t forget to recode dates)

Assignment:

• Study for Final Exam

Lesson 21: Exam 3 Review and Portfolio Check

Review:

• Exam 3 Results
• Schedule
  • This week: Exam Review, Portfolio Check (today) and Makeup Exam (Thu)
  • Next week: Thanksgiving
  • Week of Dec 2: Review for Comprehensive Final
  • Dec 10: Final Exam at 10:30a

• 

Presentation:

• Exam 3 Solutions
• Portfolio Check

Assignment:

• Study for Makeup Exam
• Or, if you don’t need the makeup, enjoy your day off on Thursday!

Review for Exam 3

Review:

• Election Activity
• Measuring Forecast Error
• Exam 3 on Thu Nov 14

Presentation:

• Review Topics:
  • Scatterplots
  • Correlation
  • Pearson Correlation Coefficient
  • Least Squares Regression
    • Sum of Squares
    • Linear Regression
    • Calculate slope and y-intercept for linear equation
    • Calculate R-Squared
    • Forecasting with a regression equation (plug in x and solve for y)
  • Cautions about Regression and Correlation
    • Extrapolation
    • Spurious correlation
    • Correlation ≠ causation
    • Lurking variables
    • Influential observations
    • Sensitivity analysis
    • Demonstrating causation
      • Retrospective studies
      • Prospective studies

Lesson 20: Measuring Forecast Error

Review:

• Election forecasting
• Next week
  • Review for Exam 3 on Tue, Nov 12
  • Exam 3 on Thu, Nov 14

Presentation:

• Election exercise: plot forecast vs actual
• Measurement of forecast error
  • MAD – mean absolute deviation
    • MAD = ∑|Actual – Forecast|/(Number of Forecasts)
  • RMSE – root mean square error
    • RMSE = √(∑(Actual – Forecast)^2/(Number of Forecasts))

• What can be done if the bivariate relationship is nonlinear?
• Curve fitting
• Use Trendline function in scatter plot
  • Linear
  • Polynomial
• Demonstrate
  • Oil Prices

Activity:

• Fit polynomial curves
  • Download interest rate spread data from the St. Louis Federal Reserve
  • Fit a 2nd and 3rd order polynomial to the data
  • Display the equation and the R^2 value
• Repeat with Corn Prices
• Add to your portfolio

Lesson 19: Special Election Forecasting Exercise

Review:

• Correlation vs Causation
• Schedule
  • Week of Nov 11 – Review and Exam 3 (Thu Nov 14)
  • Week of Nov 18 – Return Exam 3 + Portfolio check + Make Up Exam (Thu Nov 21)
  • Week of Nov 25 – Thanksgiving
  • Week of Dec 2 – Review for Final Exam (Tue Dec 10 at 10:30)
• What if I don’t need to take the Make Up Exam?

Activity:

• Election Forecasting
• Here’s some helpful County Level data along with your county assignments
• Vote by County
• Identify your assigned Colorado Counties (n=8 per student)
• Generate a forecast for each assigned County
  • % Trump/Vance
  • % Harris/Walz
  • % Other
  • Turnout (total votes cast for President)
• Considerations
  • How has your assigned County changed since 2020?
  • Has there been population growth?
  • Has it become more conservative or more liberal?
  • Will voter turnout increase or decrease?
  • 2022 midterm election results
  • Recent polling, if any.
• Submit your forecasts (on index card) – Due Today!
  • Name
  • List of Counties
  • Forecasts for each County

Assignment:

• Collect the actual numbers for each assigned County
• Bring to class on Thursday
• We will measure forecast error on Thursday
• Forecast error will help determine your participation score for this assignment

Lesson 18: Cautions about Correlation and Regression

Review:

• Tuesday activity data
• Pearson Correlation Coefficient
• R-Squared
• Schedule
  • Week of Nov 4 – Residuals (and maybe some election data)
  • Week of Nov 11 – Review and Exam 3 (Thu Nov 14)
  • Week of Nov 18 – Return Exam 3, portfolio check and Make Up Exam (Thu Nov 21)
  • Week of Nov 25 – Thanksgiving
  • Week of Dec 2 – Review for Final Exam (Tue Dec 10 at 10:30)

Presentation:

• Cautions about Correlation and Regression
  • Influential Observations
    • Outliers
    • Diabetes and blood sugar (p. 129-130)
    • Use Tuesday activity data
  • Lurking Variables
    • Explanatory variables not included
    • Beyond scope
• • 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 glucose level data

Presidential Election Year	Time Period	Ballots Cast (thousands)
2004	1	68.4
2008	2	73.9
2012	3	77.7
2016	4	78.7

• 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:

Price of Coffee ($ per pound)	Deforestation (%)
0.29	0.49
0.40	1.59
0.54	1.69
0.55	1.82
0.72	3.10

• 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 scatter plot)
  • Recalculate the linear regression equation and the corresponding R-Squared value
  • Recalculate Deforestation % assuming Price of Coffee is $0.90 per pound.

• Go back to Lesson 15 and use the data from the Activity section
• Follow the same process, i.e., look for outliers, remove and recalculate.

Lesson 17: Pearson Correlation Coefficient and R-Squared

Review:

• Linear Regression
• Pueblo County 2020 voter turnout estimates

Presentation:

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

Activity:

x	y
10	-30
3	-2
5	-10
1	6
6	-14

• 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

Femur (cm)	Humerus (cm)
38	41
56	63
59	70
64	72
74	84

• 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 16: Linear Regression

Review:

• Exam 2 review
• Scatterplots
• Sum of Squares

Presentation:

• 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 the Beer Party Regression Example
  • Video: Fitting Lines to Data

Demonstrate in Sheets:

• 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 (rather than year) as the independent (x) variable
      • Use Ballots Cast as the dependent (y) variable
      • To forecast 2020 turnout, solve for y-hat when x = 5
      • To forecast 2024 turnout, solve for y-hat when x = 6

Presidential Election Year	Time Period	Ballots Cast
2004	1	68,371
2008	2	73,881
2012	3	77,671
2016	4	78,652

Assignment in Sheets:

• Download the Snowpack and Runoff in the Arkansas River Basin
• Generate a scatter plot
  • x-axis = Column B “Freemont Pass Snow H2O Equiv (in) on March 1” and
  • y-axis = Column C: Arkansas R at Pueblo Streamvolume Observed (ac_ft) in May”
• Add a Trendline and display the linear Equation
• Create a function in Sheets to estimate runoff based on snowpack
• Generate runoff estimates assuming x=5, 10, 15, 20 and 25 inches snowpack (produce 5 forecasts)

Lesson 15: Least Squares Regression

Review:

• Exam 2

• 

• Exam Solutions
• All-purpose Make-up Exam, date TBD
• 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.

Review for Exam 2

Review:

• Scatterplots
• Exam 2
  • Thu, Oct 17
  • Covers Lessons 7–13 (does not include Lesson 14 – Scatterplots)

Exam 2 Topics:

• Census
• Sampling
• Sampling Distributions
• Central Limit Theorem
• Estimation
  • Confidence Intervals = Point estimate +/- Margin of error
  • Z-table
  • t-distribution for small samples
  • Inference for Proportions
• Significance Testing
  • Step 1. Set up Ho and Ha
  • Step 2. Calculate Z statistic
  • Step 3. Calculate p-value
  • Step 4. Compare p-value to alpha (0.05) and decide to Reject Ho or Fail to Reject Ho

Activity:

• Practice Exam