CONSIDER: Choosing Where to Meet Up

Three friends - Harper, Luke, and Jaxon - are trying to decide the best spot to meet up. Compare the two options based on how far each friend would need to travel to get there.

Cafe Mezze

• Harper lives next door and is 0 miles away

• Luke lives 15 miles away

• Jaxon lives 3 miles away

Rabat Restaurant

• Harper lives 6 miles away

• Luke lives 5 miles away

• Jaxon lives 7 miles away

Residuals

In the Intro, you compared how far the three friends were from a particular place. You can apply the same concept to a scatterplot with a line of best fit. The vertical distance between a data point and the line is called a residual. Just like the friends chose the restaurant that was best for all of them, we can use residuals to find the line that best fits the data points.

Let’s look at the example scatterplot below, which shows data on 6 colleges that Kelly is interested in. Kelly looked up the acceptance rate and average students grant aid for each college. Use the graph to answer the following questions.

GRAPH: Sticker Price and Student:Faculty Ratio

When choosing a college, one factor you might consider is the student:faculty ratio. This ratio tells you how many faculty members there are compared to the number of students. A lower student: faculty ratio might mean smaller class sizes, more opportunities to talk to professors, and more individualized support

The scatterplot below shows data on sticker price and student:faculty ratio for a representative sample of 27 4-year US colleges.

Ordinary Least Squares (OLS) Regression

Residuals are a part of finding the line of best fit. That process is called Ordinary Least Squares (OLS) Regression.

OLS regression figures out which line is the BEST fit by finding the line that minimizes the sum of the squared residuals.

Let’s see how OLS regression works for our data on sticker price and student:faculty ratio. The graph on the left shows the residuals for all of the data points. The graph on the right squares each of those residuals. The best fit line makes the total area of those squares as small as possible.

Level 1

Tuition and Graduation Rate

Ethan is researching the relationship between a college’s sticker price of a college and graduation rate. He finds data for 35 4-year colleges and uses “sticker price” to represent the published annual tuition and fees for an out-of-state student.[1]

Ethan uses linear regression to find the line of best fit. Here are his results:

ŷ = 0.87x + 35.98

r^2 = 0.57

Level 2

The Costs of College Over Time

Kiana is researching how the costs of college have changed over time. She finds data on the average annual cost of attending a 4-year college since 1970, adjusted for inflation. It includes the cost of tuition, required fees, housing, and food.

Kiana uses linear regression to find the line of best fit. Here are her results:

ŷ = 0.42x + 8.11

r^2 = 0.95

Level 3

VIDEO: What is a Residual Plot?

We can learn a lot more about our model from residuals. A residual plot is a graph that shows just the residuals from a regression model. Analyzing it can help you notice patterns and determine whether you’ve chosen the best type of function to fit the data. Watch the video and answer the questions.