Chapter 10: Two Quantitative Variables (Correlation and Regression)

What is a scatterplot and what does it represent?

A scatterplot shows the relationship between two quantitative variables measured on the same individuals, using dots to represent individual pairs. X-axis: explanatory; Y-axis: response.

What are the three main aspects of a scatterplot?

1. Form (linear, curved)

2. Direction (positive, negative, or none)

3. Strength (tightness of the points to a form)

What does a positive, negative, or no linear relationship mean?

• Positive: As x increases, y increases

• Negative: As x increases, y decreases

• No linear relationship: x and y don’t follow a linear trend

 What’s the difference between “no linear” and “no relationship”?

 No linear relationship doesn’t mean no relationship at all—it might be nonlinear.

What is the correlation coefficient (r)?

A numerical summary of the strength and direction of a linear relationship between two quantitative variables.

Range: -1 ≤ r ≤ 1

 What are key properties of the correlation coefficient r?

• Same sign as slope

• Unitless

• Affected by outliers

• Only for quantitative variables

• r =1: perfect linear relationship

 Rule of Thumb for interpreting r?

What are the hypotheses for testing correlation \rho?

𝐻0: There is no linear relationship between 𝑥 and 𝑦

𝐻𝑎: There is a linear relationship between 𝑥 and 𝑦

How is the simulation approach used to test correlation?

 Shuffle the y-values to simulate the null hypothesis and generate a distribution of r-values under H_0.

How is a confidence interval for p estimated via simulation?

What is a deterministic model?

Why is a deterministic model unrealistic in most real-world data?

Because in practice, random variation (due to uncontrolled factors or measurement error) always exists, so we move to probabilistic models.

What is the least squares regression line?

A line that best fits the data by minimizing the sum of squared residuals

What do b0 and b1 represent?

• b0: y-intercept (when x = 0)

• b1: slope (change in y per unit x)

Straight-Line Linear Models

Methods of least square

What is a residual?

e = y - y(hat)

It’s the vertical distance between an actual data point and the predicted point.

What is R^2 and how is it interpreted?

 Coefficient of determination. It represents the proportion of variance in y explained by x.

Interpreting R²

Rule of thumb for R²?

What are the hypotheses for testing slope \beta_1?

What is the link between slope and correlation in regression?

Testing B1 = 0 is equivalent to testing P = 0 for linear models.

What are the validity conditions for theory-based slope inference?

1. Linearity

2. Symmetric residuals (mean zero)

3. Equal variance along x

What is the test statistic for slope in regression?

 How do we interpret the confidence interval for B1?

If 0 is not in the CI → significant slope → strong evidence for a linear relationship.

What should we be cautious of in regression?

• Extrapolation beyond data range

• Outliers that affect line

• Influential points far in x-direction that pull the regression line

What is a probabilistic model in regression?

Why must the mean of error \epsilon be zero in regression?