ap statistics section 3.2 a regression lines. linear relationships between two quantitative...
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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.TRANSCRIPT
AP Statistics Section 3.2 ARegression Lines
Linear relationships between two quantitative variables are quite common.
Correlation measures the direction and strength of these relationships. 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.
Who:______________________________What:______________________________ ______________________________Why:_______________________________When, where, how and by whom? The data come from a controlled experiment in which subjects were forced to overeat for an 8-week period. Results of the study were published in Science magazine in 1999.
16 healthy young adultsExp.-change in NEA (cal)Resp.-fat gain (kg)
Do changes in NEA explain weight gain
NEA (calories)
Fat
Gain
(kg)
-100 0 100 200 300 400 500 600 700
8
6
4
2
0
NEA (calories)
Fat
Gain
(kg)
-100 0 100 200 300 400 500 600 700
8
6
4
2
0
Numerical summary: The correlation between NEA
change and fat gain is r = _______
7786.
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 __________.
bxay ˆ
slopey-intercept
The formula at the right will allow you to find the value of b:
x
y
SS
rb
Once you have computed b, you can then find the value of a using
this equation.
)(xbya
We can also find these values on our TI-83/84.
earlierr found way wesame
For this example, the LSL is
or
xy 0034.505.3ˆ
.))((0034.505.3)( calNEAchangekgFatGain
Interpreting b: The slope b is the predicted _____________ in the
response variable y as the explanatory variable x changes.
rate of change
The slope b = -.0034 tells
us that fat gain goes down by .0034 kg for each additional
calorie of NEA.
You cannot say how important a relationship is by looking at how
big the regression slope is.
Interpreting a: The y-intercept a = 3.505 kg is the fat gain estimated by the model if
NEA does not change when a person overeats.
Model: Using the equation above, draw the LSL on your scatterplot.
NEA (calories)
Fat
Gain
(kg)
-100 0 100 200 300 400 500 600 700
8
6
4
2
0
5007.1
10034.
10000340034.
TI 83/84 8:LinReg(a+bx)
GRAPH
121 ,, YLL
ENTERYFunctionVARSY
VARS
1:1:1
Prediction: Predict the fat gain for an individual whose NEA increases
by 400 cal by:
(a) using the graph ___________
(b) using the equation _________
NEA (calories)
Fat
Gain
(kg)
-100 0 100 200 300 400 500 600 700
8
6
4
2
0
Prediction: Predict the fat gain for an individual whose NEA increases
by 400 cal by:
(a) using the graph ___________
(b) using the equation _________
2.2
)400(0034.505.3ˆ y
Prediction: Predict the fat gain for an individual whose NEA increases
by 400 cal by:
(a) using the graph ___________
(b) using the equation _________
2.2
145.2
Predict the fat gain for an individual whose NEA increases by
1500 cal.
595.1ˆ)1500(0034.505.3ˆ
yy
So we are predicting that this individual loses fat when he/she
overeats. What went wrong?
1500 is way outside the range of NEA values in our data
Extrapolation is the use of a regression line for prediction
outside the range of values of the explanatory variable x used to
obtain the line. Such predictions are often not accurate.
ab