Download 3.2a notes A regression line is a line that describes how a response

3.2a notes
A regression line is a line that describes how a response variable y changes as an explanatory
variable x changes. We often use a regression line to predict the value of y for a given value of
x.
𝑦̂ = 𝑎 + 𝑏𝑥
where 𝑦̂ (y hat) is the predicted value, b is a slope, and a is the y intercept.
Example 1: Used Hondas
The following data shows the number of miles driven and advertised price for 11 used Honda
CR-Vs from the 2002-2006 model years (prices found at www.carmax.com). The scatterplot
below shows a strong, negative linear association between number of miles and advertised cost.
The correlation is -0.874. The line on the plot is the regression line for predicting advertised
price based on number of miles.
Thousand
Miles
Driven
22
29
35
39
45
49
55
56
69
70
86
Cost
(dollars)
17998
16450
14998
13998
14599
14988
13599
14599
11998
14450
10998
Example 2: The regression line shown on the scatterplot is ŷ = 18773 – 86.18x or
price  18773 – 86.18(number of miles) .
3.2a notes
Identify the slope and y intercept of the regression line. Interpret each value in context.
Extrapolation: is the use of a regression line for prediction far outside the interval of values of
the explanatory variable x used to obtain the line. Such predications are often not accurate.
Example 3: Used Hondas
For the advertised price and number of miles data, the equation of the regression line is:
price  18,773 – 86.18(thousands of miles) . Predict the asking price for a car with 50,000
miles. Substitute x = 50 into the regression line: price  18,773 – 86.18(50) = $14,464. This is
illustrated in the figure below.
Example 4: Extrapolation and Used Hondas
Should we predict the asking price for a used 2002-2006 Honda CR-V with 250,000 miles? No!
We only have data for cars with between 22,000 and 86,000 miles. We don’t know if the linear
pattern will continue beyond these values. In fact, if we did predict the asking price for a car
with 250 thousand miles, it would be −$2772!