## Lesson 15: Least Squares Regression

October 22, 2024

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.*