Lesson 18: Linear Regression

April 6, 2026

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