AP Statistics Section 3.2 A Regression Lines

Linear relationships between two quantitative variables are quite common. Just as we drew a density curve to model the data in a histogram, we can summarize the overall pattern in a linear relationship by drawing a _______________ on the scatterplot. regression line

Note that regression requires that we have an explanatory variable and a response variable. A regression line is often used to predict the value of y for a given value of x.

A least-squares regression line relating y to x has an equation of the form ___________ In this equation, b is the _____, and a is the __________. slope y-intercept

NOTE: You must always define the variables (i.e. and x) in your regression equation.

The formulas below allow you to find the value of b depending on the information given in the problem:

Once you have computed b, you can then find the value of a using this equation.

TI-83/84: Do the exact same steps involved in finding the correlation coefficient, r.

Example 1: Let’s revisit the data from section 3.1A on sparrowhawk colonies and find the regression equation.

Interpreting b: The slope b is the predicted _____________ in the response variable y as the explanatory variable x increases by 1. rate of change

Example 2: Interpret the slope of the regression equation for the data on sparrowhawk colonies.

You cannot say how important a relationship is by looking at how big the regression slope is.

Interpreting a: The y-intercept a is the value of the response variable when the explanatory variable is equal to ____.

Example 3: Interpret the y- intercept of the regression equation for the data on sparrowhawk colonies.

Example 4: Use your regression equation for the data on sparrowhawk colonies to predict the number of new birds coming to the colony if 87% of the birds from the previous year return.

CAUTION: Extrapolation is the use of a regression line for prediction outside the range of values of the explanatory variable used to obtain the line. Such predictions are often not accurate.

Example 5: Does fidgeting keep you slim? Some people don’t gain weight even when they overeat. Perhaps fidgeting and other non- exercise activity (NEA) explains why - some people may spontaneously increase NEA when fed more. Researchers deliberately overfed 16 healthy young adults for 8 weeks. They measured fat gain (in kg) and change in energy use (in calories) from activity other than deliberate exercise.

Construct a scatterplot and describe what you see.

Write the regression equation and interpret both the slope and the y-intercept.

Predict the fat gain for an individual whose NEA increases by 1500 cal.

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