Lesson 20: Cautions about Correlation and Regression

Review:

• Pearson Correlation Coefficient
• R-Squared
• Schedule
  • Finish Simple Regression this week
  • Review and Exam 3 next week

Presentation:

• Cautions about Correlation and Regression
  • Influential Observations
    • Outliers
  • Lurking Variables
    • Explanatory variables not included
    • Beyond scope
  • 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
  • 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 study = following a sample population over time and studying behaviors possibly linked to likelihood of an outcome
  • Video

Demonstration in Sheets:

Last week we gathered data 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. Today, let’s look at the data in Sheets to produce the regression equation and point estimate for 2026

Activity/Assignment:

• Use Sheets (or Excel) to solve the problems from last week (see below)
• Use Sheets to find the regression equation for this larger real estate data set

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 19: Pearson Correlation Coefficient and R-Squared

Review:

• Linear Regression

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 a new problem from beginning to end.

Demonstration Problem. A street vendor sells hot chocolates on an outdoor pedestrian mall. Sales tend to be strong on cold days but weak on warmer days. Calculate the Sum of Squares, Linear Regression, Pearson Correlation Coefficient and R-Squared.

Day	Temp (x)	Hot Chocolates Sold (y)
1	30	85
2	35	78
3	40	82
4	45	70
5	50	74
6	55	66
7	60	63
8	65	68

Use the resulting y^ equation, estimate hot chocolate sales assuming 32 degrees and 64 degrees, respectively.

How does this compare to the linear model for estimating Iced Coffee sales? Which would be more accurate for sales forecasting? Why?

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
2025	18.2	817.9

Calculate Sum of Squares, the Linear Equation, the Pearson Correlation Coefficient and R^2. If it helps, you may divide the streamflow numbers by 100 so the squared values aren’t too ridiculously large.

Using the regression equation, estimate Streamflow for 2026 assuming Snowpack is only 8.6 (this year is the worst year for Snowpack in Colorado history).

Lesson 18: Linear Regression

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: Lemonade Stand
  • Use temperature to predict sales
  • Sample data
    • Day 1: 70°, 45
    • Day 2: 75°, 52
    • Day 3: 80°, 54
    • Day 4: 85°, 60
  • Scatterplot
  • Table setup
    • columns: x, y, x^2, y^2, xy
    • sums: ∑x, ∑y, ∑x^2, ∑y^2, ∑xy
  • Sum of Squares calculations
  • Linear Equation (slope + intercept)
    • b1 = SSxy/SSxx
    • b0 = (∑y/n) – b1*(∑x/n)
• Example 2: Iced Coffee Sales by Temp
  • Use temperature to predict sales
  • Sample data
    • Day 1: 58°, 40
    • Day 2: 62°, 44
    • Day 3: 68°, 49
    • Day 4: 72°, 53
    • Day 5: 78°, 59
    • Day 6: 82°, 63
    • Day 7: 88°, 67
    • Day 8: 92°, 73
  • Scatter Plot
  • Table setup
    • columns: x, y, x^2, y^2, xy
    • sums: ∑x, ∑y, ∑x^2, ∑y^2, ∑xy
  • Sum of Squares calculations
    • SSxx = ∑x² – (∑x)²/n
    • SSyy = ∑y² – (∑y)²/n
    • SSxy = ∑xy – (∑x*∑y)/n
• Video: Fitting Lines to Data

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 food truck business tracks daily high temperature (°F) and the number of customers. Calculate Sum of Squares and the equation of the regression line.

Day	Temp (x)	Customers (y)
1	70	45
2	75	48
3	80	52
4	85	60
5	90	57

Problem 2. What is the relationship between vehicle age and resale value on the used car market?

Vehicle	Age (x, Years)	Resale Value (y, $1000s)
1	2	26
2	4	22
3	6	21
4	8	17
5	10	19
6	12	15

Problem 3. A property manager wants to estimate the relationship between unit square footage (x) and rent (y). Calculate Sum of Squares and the equation of the regression line.

Unit	Sq Ft (x)	Rent (y, $)
A	800	1300
B	900	1280
C	1000	1450
D	1100	1500
E	1200	1400

Problem 4. A delivery company tracks delivery time (x) and customer satisfaction ratings (y, 1–10 scale). Calculate Sum of Squares and the equation of the regression line.

Delivery	Time (x, minutes)	Satisfaction (y)

A

20

9

B

30

8

C

40

6

D

50

7

E

60

5

Lesson 17: 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: Lemonade Stand
    • Use temperature to predict sales
    • Sample data
      • Day 1: 70°, 45
      • Day 2: 75°, 52
      • Day 3: 80°, 54
      • Day 4: 85°, 60
    • Scatterplot
    • Table setup
      • columns: x, y, x^2, y^2, xy
      • sums: ∑x, ∑y, ∑x^2, ∑y^2, ∑xy
    • Sum of Squares calculations
      • SSxx = ∑x² – (∑x)²/n
      • SSyy = ∑y² – (∑y)²/n
      • SSxy = ∑xy – (∑x*∑y)/n
  • Example 2: Iced Coffee Sales by Temp
    • Use temperature to predict sales
    • Sample data
      • Day 1: 58°, 40
      • Day 2: 62°, 44
      • Day 3: 68°, 49
      • Day 4: 72°, 53
      • Day 5: 78°, 59
      • Day 6: 82°, 63
      • Day 7: 88°, 67
      • Day 8: 92°, 73
    • Scatter Plot
    • Table setup
      • columns: x, y, x^2, y^2, xy
      • sums: ∑x, ∑y, ∑x^2, ∑y^2, ∑xy
    • Sum of Squares calculations
      • SSxx = ∑x² – (∑x)²/n
      • SSyy = ∑y² – (∑y)²/n
      • SSxy = ∑xy – (∑x*∑y)/n
  • We will use these Sum of Squares calculations (next time) to determine
    • linear equation y = ax + b
    • 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 16: 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 – 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: Make a Scatterplot to visualize the relationship between Advertising and Sales.

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: Make a Scatterplot to visualize the relationship between Training Hours and Productivity.

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: Make a Scatterplot to visualize the relationship between Price and Units Sold.

Problem 4. 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 5. 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 4-5 copied from the Against All Odds video series guide for Unit 10.