Lesson 20: Pearson Correlation Coefficient and R-Squared

Review:

• Linear Regression
• Exam 2 corrections
  • Only for test takers who took Exam 2 on the scheduled date.
  • You must complete this entire exam.
  • Your exam score will increase up to 10 points or up to 50% of points deducted, whichever is less. For example, if you received an 88 on the exam you can only earn up to 6 points.
  • Your actual points awarded will be based on your exam score. For example, if you get an 80% on the new exam, you will earn 8/10 points (or .8 x 50% of points deducted).

Presentation:

• 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
• Let’s solve this problem from last time, including calculating r and r^2, from beginning to end.

Problem 2. A local brewery tracks weekly social media ad spending and kegs sold. Calculate Sum of Squares and the equation of the regression line.

Week	Ad Spend (x, $100’s)	Kegs Sold (y)
1	2	11
2	3	15
3	4	17
4	5	20
5	7	26
6	9	30

Let’s also estimate kegs sold when the ad spend is $650 (x=6.5)

Video: Fitting Lines to Data

Activity:

Now, here’s some real data, gathered from USDA National Water and Climate Center and the USGS Water Data Center. The snowpack data (snow-water equivalent inches) is for the Freemont Pass location. The streamflow data (ft^3/SEC) is for the Arkansas River station in Pueblo.

Water Year	Snowpack (x)	Streamflow (y)
2018	20.4	577.5
2019	28.1	1599.3
2020	20.2	901.0
2021	15.5	659.4
2022	17.6	730.4
2023	17.1	1016.0
2024	22.2	1455.7

Calculate Sum of Squares, the Linear Equation, the Pearson Correlation Coefficient and R^2.

Using the regression equation, estimate Streamflow for 2025 assuming Snowpack was 18.2.

Assignment:

• Complete the corrections exam or enjoy a pleasant weekend.

Lesson 19: 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 1: Beer Sales by Temp
  • Use temperature to predict kegs sold
  • Historical Data (Temp degrees F, # of kegs sold)
    • Day 1: 60°, 12
    • Day 2: 70°, 14
    • Day 3: 80°, 16
    • Day 4: 90°, 18
    • Day 5: 100°, 20
  • Scatter Plot
  • Table setup
• Sum of Squares calculations
  • SSxx = ∑x² – (∑x)²/n
  • SSyy = ∑y² – (∑y)²/n
  • SSxy = ∑xy – (∑x*∑y)/n
• Calculate Slope of regression line
  • b1 = SSxy/SSxx
• Calculate the y-intercept
  • b0 = (∑y/n) – b1*(∑x/n)
• Video: Fitting Lines to Data

Activity:

• Example 2: Beer Sales by Price
  • Historical Data (Price $ # of kegs sold)
    • Day 1: $90, 22
    • Day 2: $100, 20
    • Day 3: $110, 18
    • Day 4: $120, 16 
    • Day 5: $130, 14
  • Calculate Sum of Squares
  • Calculate the slope and y-intercept
  • Write the linear equation

Assignment:

First, go back to the problems assigned last week. Use your Sum of Squares calculations to find the equation of the regression line for each problem.

1. Advertising vs. Sales (n = 8)
   • SSxx = 55.5
   • SSyy = 1,385.875
   • SSxy = 275.75
2. Training Hours vs. Productivity (n = 8)
   • SSxx = 42
   • SSyy = 327.5
   • SSxy = 117
3. Price vs. Units Sold (n = 8)
   • SSxx = 1050
   • SSyy = 29,400
   • SSxy = –5,500

Problem 1. A small coffee shop tracks daily high temperature (°F) and the number of iced coffees sold. Calculate Sum of Squares and the equation of the regression line.

Day	Temp (x)	Iced Coffees Sold (y)
1	60	42
2	65	48
3	70	52
4	75	61
5	80	65

Problem 2. A local brewery tracks weekly social media ad spending and kegs sold. Calculate Sum of Squares and the equation of the regression line.

Week	Ad Spend (x, $100’s)	Kegs Sold (y)
1	2	11
2	3	15
3	4	17
4	5	20
5	7	26
6	9	30

Problem 3. A realtor wants to estimate the relationship between square footage (x) and listing price (y in $1000s). Calculate Sum of Squares and the equation of the regression line.

House	Sq Ft (x)	Listing Price (y)
A	1,600	315
B	1,850	344
C	2,000	365
D	2,300	399
E	2,500	425

Lesson 9: Intro to Monte Carlo Simulation

Review:

• Multiple Regression exam results

• 

• Exam solutions
• Used car pricing model submissions

Presentation:

• Probability
  • Topic worthy of its own course
• Simulation
  • Random phenomenon
    • Individual outcomes are uncertain
    • Regular distribution of outcomes in large number of trials
  • Random number generation and stochastic processes
  • Discrete and Continuous (we’ll start with discrete)
• Coin Toss Simulation
  • Demonstrate Coin Toss Simulation in Sheets
• Dice Roll Simulation
  • Demonstrate Dice Roll Simulation in Sheets in Sheets
  • Lucky 7 game
    • Roll a 7, win bet
    • Roll anything else (other than a 7), lose bet

Assignment:

• Replicate the Coin Toss simulation
  • 200 coin tosses (instead of 32)
  • reproduce the graph (showing 200 tosses)
• Create a “Roll 2 Dice” simulation
  • 100 rolls (instead of 32)
  • Roll 2 Dice instead of only 1
  • Be sure the probability distribution is the same as actually rolling 2 Dice.
• Create a “Lucky 7” simulation
  • Roll 2 Dice
  • If you roll a 7, you win
  • If you don’t roll a 7, you lose
  • Track the number of wins and losses
• Create a “Simple Craps” simulation
  • Roll 2 Dice
  • If you roll a 7 or 11, you win $10
  • If you roll a 2, 3 or 12, you lose $5
  • If you roll anything else (i.e., 4, 5, 6, 8, 9, 10), it’s a “push” or a tie and you don’t lose or win $$.
  • Set up a scoring system to keep track of how much $$ you win or lose.

Lesson 18: 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 1: Beer Sales by Temp
    • Use temperature to predict kegs sold
    • Historical Data (Temp degrees F, # of kegs sold)
      • Day 1: 60°, 12
      • Day 2: 70°, 14
      • Day 3: 80°, 16
      • Day 4: 90°, 18 
      • Day 5: 100°, 20
    • Scatter Plot
    • Table setup
  • Sum of Squares calculations
    • SSxx = ∑x² – (∑x)²/n
    • SSyy = ∑y² – (∑y)²/n
    • SSxy = ∑xy – (∑x*∑y)/n
  • Example 2: Beer Sales by Price
    • Experiment with price to predict demand
    • Historical Data (Price $ # of kegs sold)
      • Day 1: $90, 22
      • Day 2: $100, 20
      • Day 3: $110, 18
      • Day 4: $120, 16 
      • Day 5: $130, 14
    • Students complete calculations
  • We will use these Sum of Squares calculations (next time) to determine
    • linear equation
    • Pearson correlation coefficient
    • R-squared

Activity:

Problem 1 – Advertising and Sales (n = 8)

A retail chain studied the link between weekly advertising spend and sales (in $1,000s).

Week	Advertising (x)	Sales (y)
1	6	64
2	8	77
3	10	85
4	4	55
5	7	72
6	9	82
7	3	50
8	11	88

Task:
Calculate SSxx, SSyy and SSxy

Problem 2 – Training Hours and Productivity (n = 8)

An HR analyst examined whether more employee training hours lead to higher productivity (daily packages processed per worker).

Dept	Training Hours (x)	Productivity (y)
A	5	46
B	8	54
C	4	42
D	6	49
E	10	59
F	3	40
G	9	57
H	7	51

Task:
Calculate SSxx, SSyy and SSxy

Problem 3 – Price and Units Sold (n = 8)

A manufacturer tested different price points to estimate demand.

Trial	Price (x)	Units Sold (y)
1	40	210
2	35	250
3	45	190
4	50	160
5	30	280
6	55	150
7	25	310
8	60	130

Task:
Calculate SSxx, SSyy and SSxy

Lesson 17: Scatterplots

Review:

• Exam 2 Results

• 

Presentation:

• Scatterplots
  • Visualize relationship between 2 variables
    • independent (x) and dependent (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
  • Unit 10 Scatterplots Video

Activity:

Complete the problems below twice, once by hand and once in Sheets or Excel.

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.