• The (population) regression line: \(y = \beta_0 + \beta_1 x + \epsilon\)
• When \(\beta_1=0 \Longrightarrow\) no relationship between variables x and y
• The p-value for \(\beta_1\) is based on:
  • \(H_0: \beta_1=0\) vs
  • \(H_A: \beta_1\neq 0\)
• If CI for \(\beta_1\) does not contain 0, it suggests a significant relationship between x and y.

What does "best fitting line"" mean?

Consider ANY point (in blue).

Now consider this point's deviation from the regression line.

Do this for another point…

Do this for another point…

Regression line minimizes the sum of squared arrow lengths.

• Population regression line: \[y = \beta_0 + \beta_1 x + \epsilon\]
• Sample AKA fitted regression line based on points \((x_i, y_i)\) for \(i=1, \ldots, n\): \[ \widehat{y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 x_i \]

• \(y_i\) observed value: blue point
• \(\widehat{y}_i\) fitted value: value on red line
• Residual \(\epsilon_i = y_i - \widehat{y}_i\): length of arrow

• They're the error, the lack of fit, what's leftover.
• It's the noise around the signal (regression line).
• Punchline: We want there to be no systematic pattern in the residuals.
• Kind of a big deal…