📊 Part 1: Understanding Gradients

1. A linear regression line has the equation y = 2.5x + 10, where x represents hours studied and y represents test scores. What does the gradient tell us?

Test scores increase by 2.5 points for every additional hour studied

Students must study for 2.5 hours to pass

The maximum test score is 2.5 points

Test scores decrease by 2.5 points per hour

2. Which of these gradients indicates the strongest negative relationship?

m = -0.3

m = -1.8

m = 0.7

m = -2.4

3. A regression line shows the relationship between temperature (°C) and ice cream sales ($). The gradient is 15. This means:

For every 1°C increase in temperature, ice cream sales increase by $15

Ice cream costs $15

Temperature increases by 15°C when sales increase

Sales decrease by $15 for every degree increase

4. Look at this regression equation: Height = -0.8 × Age + 180 (for tree height in cm and age in years). The negative gradient suggests:

Trees grow taller as they age

Trees shrink by 0.8 cm each year

This relationship doesn't make sense for tree growth

The tree starts at 180 cm tall

✏️ Part 2: Interpreting Real-World Data

5. A study shows the relationship between weekly exercise hours (x) and resting heart rate (y) with the equation: y = -2.3x + 75

a) What is the gradient? _______

b) Interpret the gradient in context:

c) What would be the predicted resting heart rate for someone who exercises 4 hours per week?

6. A linear regression analysis of car age (years) vs car value ($) gives: Value = -2800 × Age + 25000

a) What does the gradient of -2800 represent in this context?

b) Is this gradient realistic? Explain your reasoning:

7. A researcher finds that plant growth (cm) relates to rainfall (mm) with a gradient of 0.05. Explain what this means and discuss one limitation of using this model for predictions:

🤔 Part 3: Critical Thinking

8. Consider the ethical implications: A school uses a linear regression model (gradient = 0.6) to predict student achievement based on family income. What concerns might arise from using this model?

9. Extension Task: Create your own example of a bivariate relationship. Write the regression equation and explain what the gradient means in your context:

Variables: _________________ and _________________

Equation: y = _______ x + _______

Gradient interpretation: