Lesson 17: Least Squares Regression

April 1, 2026

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