Today you will investigate residuals. A
residual is the difference between what
actually occurred and what was
predicted by your best-fit model.
6-10. Battle Creek Cereal is trying a variety of packaging
sizes for their Crispy Puffs cereal. Below is a list of six
current packages.
• Write a few sentences to the
executives of Battle Creek
Cereal that describe the
association, and give an
equation the executives can use
to predict the net weight of
cereal based on the amount of
cardboard used for the
package. Tell the executives
how much cereal a new
experimental “green” package
that uses 260 in2 of cardboard
is expected to hold.
6-11. A residual is a measure of how far a prediction is from
what is actually observed.
residual = actual − predicted
The 260 in2 box from problem 6-10 will actually hold 355 g of
cereal.
• Your Task: Discuss with
your team how you can
indicate a residual as a
distance on your
scatterplot from problem
6-10. Be prepared to
share what you discuss
with the class.
• What is the residual for
the 260 in2 box?
• What is the difference
between a positive and a
negative residual in the
context of this problem?
• How could graphing the
actual and predicted data
points help?
6-12
• What is the residual for the 471 in2 box? Mark
the residual on your scatterplot. Be sure to
include units for your residual.
6-13. The warehouse store wants to
offer a super-sized 600 in2 box.
a) The residual for this box is 1005 grams. What
is the actual weight of a 600 in2 box?
b) Why do you suppose the residual is so
large?
c) Interpret the meaning of the slope and yintercept of your model in the context of this
problem. Does the y-intercept make sense in
the context of the problem?
6-14. Armen was concerned about the amount of sugar in his
diet, so he went to the store and collected data from several
cereal boxes. Armen used the data to create a model that
related the sugar in cereal to calories: s = –16.9 + 0.23c where s
is the amount of sugar in grams and c is the number of calories
in one cup of cereal.
a) What does a negative residual mean in this
context? Is a cereal with a positive or
negative residual better for Armen’s diet?
b) Interpret the meaning of the slope and yintercept in the context of the
problem. Does the y-intercept make sense in
the context of the problem?
6-15. LEARNING LOG
• Work with your team to summarize what you
have discovered about residuals today. When
you have come to an agreement, write your
ideas as a Learning Log entry. Title this entry
“Residuals” and label it with today’s date.
• The slope of a linear association plays the same role as the
slope of a line in algebra. Slope is the amount of change we
expect in the dependent variable (Δy) when we change the
independent variable (Δx) by one unit. When describing
the slope of a line of best fit, always acknowledge that you
are making a prediction, as opposed to knowing the truth,
by using words like “predict,” “expect,” or “estimate.”
• The y-intercept of an association is the same as in
algebra. It is the predicted value of the dependent
variable when the independent variable is zero. Be
careful. In statistical scatterplots, the vertical axis is often
not drawn at the origin, so the y-intercept can be
someplace other than where the line of best fit crosses the
vertical axis in a scatterplot.
• Also be careful about extrapolating the data too
far—making predictions that are far to the right
or left of the data. The models we create can be
valid within the range of the data, but the farther
you go outside this range, the less reliable the
predictions become.
• When describing a linear association, you can use
the slope, whether it is positive or negative, and
its interpretation in context, to describe the
direction of the association.