1. Review linear equations.
2. Motivate regression.
3. Interpret a slope and intercept in context.
4. Use a regression line to predict values of a dependent variable.

• Should already have seen linear equations like \[y=mx+b\]
• In statistics, we write these as \[y=b_0 + b_1x\]
  • \(b_0\) and \(b_1\) are constants.
  • \(x\) is the independent variable.
  • \(y\) is the dependent variable

\[y=b_0 + b_1x\]

• The y-intercept is \(b_0\), the value of \(y\) when \(x=0\).
• The slope is \(b_1\), the change in \(y\) for a 1-unit change in \(x\).

A scatterplot shows the relationship between two (numeric) variables.

We call this type of data bivariate data.

This relationship can be modeled perfectly with a straight line: \[y = 8 + 3.25x\]

\[y = 8 + 3.25x\] where \(x\) is the number of coffees purchased in a month and \(y\) is the amount of money spent on coffee.

• The intercept is 8, which is the dollar amount of money spent on coffee (the value of \(y\)) when 0 coffees are purchased in that month (when \(x=0\)).
• The slope is 3.25, which is the increase in amount of money spent on coffee (increase in \(y\)) for each additional coffee purchased (a one-unit increase in \(x\)).

• But what if that pound of coffee didn’t always cost $8?
• Or the coffee drinks didn’t always cost $3.25?

The linear regression line looks like \[y = \beta_0 + \beta_1x + \epsilon\]

• \(\beta\) is the Greek letter “beta”.
• \(\beta_0\) and \(\beta_1\) are constants.
• Error (the fact that the points don’t all line up perfectly) is represented by \(\epsilon\).

We estimate \(\beta_0\) and \(\beta_1\) using data and denote the estimated line by \[\hat{y} = b_0 + b_1x \]

• \(\hat{y}\), “y-hat”, is the estimated value of \(y\).
• \(b_0\) is the estimate for \(\beta_0\).
• \(b_1\) is the estimate for \(\beta_1\).
• …and 0 is the estimate for \(\epsilon\) (so we ignore it).

We use a regression line to make predictions about \(y\) using values of \(x\).

Think of this as the 2-dimensional version of a point estimate!

• \(y\) is the response variable.
• \(x\) is the predictor variable.

Example: Researchers took a variety of measurements on 344 adult penguins in Antarctica.

The regression model for these data is \[\hat{y}=-5780.83 + 49.69x\]

To predict the body mass for a penguin with a flipper length of 180mm, we just need to plug in 180 for flipper length (\(x\)): \[\hat{y}=-5780.83 + 49.69\times 180 = 3163.37\text{g}.\] - Note that the regression line automatically deals with units.

Section 3.1 Exercises 1 and 2