Lesson 19: Linear Regression

November 4, 2025

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