ap statistics section 3.2 a regression lines
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AP Statistics Section 3.2 A Regression Lines. - PowerPoint PPT PresentationTRANSCRIPT
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AP Statistics Section 3.2 ARegression Lines
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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
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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.
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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
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NOTE: You must always define the variables (i.e. and x) in your
regression equation.y
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The formulas below allow you to find the value of b depending on the
information given in the problem:
x
y
S
Srb
2xx
yyxxb
i
ii
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Once you have computed b, you can then find the value of a using
this equation.
)(xbya
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TI-83/84: Do the exact same steps involved in finding the correlation
coefficient, r.
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Example 1: Let’s revisit the data from section 3.1A on sparrowhawk
colonies and find the regression equation.
returning) birds of .304(%-31.934 birds new #
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Interpreting b: The slope b is the predicted _____________ in the
response variable y as the explanatory variable x increases by
1.
rate of change
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Example 2: Interpret the slope of the regression equation for the data on sparrowhawk colonies.
.304by decreases birds new ofnumber
predicted theyear,next colony the the toreturning
birdsadult ofnumber in the 1% of increaseeach For
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You cannot say how important a relationship is by looking at how
big the regression slope is.
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Interpreting a: The y-intercept a is the value of the response variable when the explanatory variable is
equal to ____.0
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Example 3: Interpret the y-intercept of the regression equation for the data on sparrowhawk colonies.
31.934. iscolony in the birds new ofnumber
predicted the0, is returning birds ofpercent When the
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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.
486.5ˆ
)87(304.934.31ˆ
y
y
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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.
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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.
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Construct a scatterplot and describe what you see.
gain.fat andNEA in change the
between iprelationshlinear negative strongfairly a is There
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Write the regression equation and interpret both the slope and the y-intercept.
)change (0034.505.3gainFat NEA
.0034kgby decreasesgain fat
predicted NEA thein calorie 1 of increaseeach For
3.505kg. isgain
fat predicted theNEA,in change no is When there
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Predict the fat gain for an individual whose NEA increases by
1500 cal.
595.1ˆ
)1500(0034.505.3ˆ
y
y
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ab