Lecture Notes on Exponential and Logarithmic Regression (Section 5.4)

Overview

• Focus on exponential and logarithmic regression.
• Similar process as other regression types:
  • Enter data into lists.
  • Use a specific formula to fit the data.

Exponential Regression

• Formula: $y = a \cdot b^x$
  • $a$ and $b$ are constants determined by the calculator after data entry.
• Steps in Calculator:
  1. Press STAT, go to EDIT to enter data.
  • List 1: x-values
  • List 2: y-values
  2. Go to STAT ➔ CALCULATE ➔ 0 (Exponential Regression)
  3. Plug the resulting $a$ and $b$ into the formula.
• Example:
  • x: -2, -1, 0, 1, 2
  • y: 2, 3, 4.5, 6.75, 10.125
  • Outputs: $a = 4.5$, $b = 1.5$
  • Formula: $y = 4.5 \cdot 1.5^x$

Logarithmic Regression

• Formula: $y = a + b \cdot \ln(x)$
• Steps in Calculator:
  1. Enter data in STAT ➔ EDIT.
  • List 1: x-values
  • List 2: y-values
  2. Go to STAT ➔ CALCULATE ➔ 9 (Logarithmic Regression)
  3. Plug the resulting $a$ and $b$ into the formula.
• Example:
  • x: 1, 2, 3, 4, 5
  • y: 1.3, 4.9, 8, 9.2
  • Outputs: $a = 1.39$, $b = 4.86$
  • Formula: $y = 1.39 + 4.86 \cdot \ln(x)$

Scatter Plot and Regression Type

• Creating a Scatter Plot:
  1. Enter data into List 1 and List 2.
  2. STAT ➔ EDIT to clear and input data.
  3. Press Y=, turn Plot 1 on, and press ZOOM ➔ 9 for ZoomStat.
• Observation:
  • Check whether the plot suggests a linear or exponential fit.
  • Example indicates exponential fit due to upward curve.

Power Function

• Definition: A type of regression that models data with $y = a \cdot x^b$.
• Steps in Calculator:
  1. Enter data in List 1 and List 2.
  2. Use STAT ➔ CALCULATE ➔ PWR (Power Regression)
  3. Outputs: $a$ and $b$ must be round off to three places.
• Example:
  • Outputs: $a = 3.755$, $b = 0.499$
  • Formula: $y = 3.755 \cdot x^{0.499}$

Quadratic Function

• Formula: $y = ax^2 + bx + c$
• Steps in Calculator:
  1. Use STAT ➔ CALCULATE ➔ 5 (Quadratic Regression)
  2. Outputs are rounded to three places for $a$, $b$, and $c$.
• Example:
  • Outputs: $a = -0.152$, $b = 2.051$, $c = 1.89$
  • Formula: $y = -0.152x^2 + 2.051x + 1.89$

Additional Logarithmic Regression Example

• Steps in Calculator:
  1. Use STAT ➔ CALCULATE ➔ 9 (Logarithmic Regression)
  2. Outputs are rounded to three places.
• Example:
  • Outputs: $a = 3.600$, $b = 2.888$
  • Formula: $y = 3.600 + 2.888 \cdot \ln(x)$

Homework

• Complete MyMathLab assignment for Section 5.4.
• Practice using the calculator for different regression types.