Lesson 18: Least Squares Regression

October 30, 2025

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