Objectives:
1. To use a graphing
utility to perform a
linear regression and
use it to make a
prediction
2. Write a mathematical
model involving
direct, inverse, or
joint variation
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Assignment:
P. 110-1: 7a, c, e;
9 a-e; 10 a-e
P. 111: 25, 31
P. 112: 39-48 S
P. 112: 49-54 S
P. 113: 55-62 S, 64, 70
P. 114: 78
HW Supplement
Correlation
Positive Correlation
Negative Correlation
No Correlation
Correlation Coefficient
Line of Best Fit
Least Squares Regression
Quadratic Regression
Inverse Variation
Linear Regression
Direct Variation
Joint Variation
Let’s say a set of data
consists of two quantities,
x and y. In statistics, a
correlation exists
between x and y if there is
a linear relation between
x and y.
– If y increases as x
increases, there is a
positive correlation.
Let’s say a set of data
consists of two quantities,
x and y. In statistics, a
correlation exists
between x and y if there is
a linear relation between
x and y.
– If y decreases as x
increases, there is a
negative correlation.
Let’s say a set of data
consists of two quantities,
x and y. In statistics, a
correlation exists
between x and y if there is
a linear relation between
x and y.
– If there is no obvious
pattern, there is approx.
no correlation.
Describe the correlation shown by each scatter plot.
A correlation coefficient for a set of data
measures the strength of the correlation.
-1 ≤ r ≤ 1
• Perfect negative correlation = -1
• No correlation = 0
• Perfect positive correlation = 1
A correlation coefficient for a set of data
measures the strength of the correlation.
-1 ≤ r ≤ 1
Here are some steps to commit to memory:
1. Press STAT.
2. Under the EDIT menu, choose Edit…
3. Enter your x-values under L1.
4. Enter your y-values under L2.
5. Press 2nd MODE. This QUITS the edit screen.
6. Press 2nd Y=. This is the STAT PLOT menu.
7. Choose Plot1. Highlight On.
–
Scatter Plot, x-values = L1, y-values = L2.
8. Press ZOOM and choose ZoomStat.
The table shows the number y (in thousands) of
alternative-fueled vehicles in use in the United
States x years after 1997. Graph this data as a
scatter plot. Determine if a correlation exists.
If a strong correlation
exists between x
and y, where | r | is
near 1, then the
data can be
reasonably
modeled by a trend
line.
This line of best fit
lies as close as
possible to all the
data points, with as
many above as
below.
More steps to memorize:
1. Press STAT.
2. Under the CALC menu, choose the appropriate
regression.
3. Press L1 (2nd 1).
4. Press , (comma).
5. Press L2 (2nd 2).
6. Press , (comma).
7. Press VARS.
8. Under the Y-VARS menu, choose Function…, then Y1.
Find the best-fitting line from Exercise 2.
Interpret the meaning of the slope and yintercept of your equation. Use your model to
predict y when x = 10.
The table gives the systolic blood pressure y of
patients x years old. Determine if a
correlation exists. If it is a strong correlation,
find the line of best fit. Then predict the
systolic blood pressure of a 16-year-old.
•
According to your model
Y(16)=96, but:
Girls (Age 16)
Boys (Age 16)
122-132
125-138
• Our prediction was so far
off because you can’t
reliably extrapolate this
far away from the data
What your calculator is
computing when it
finds a line of best fit
is the least squares
regression.
Looking at the image, this
occurs where the
square of the
difference between the
actual data and the
predicted data is the
least.
Regression can also be
performed on a
nonlinear set of data.
Quadratic Regression is the process of finding a
best-fitting quadratic model for a set of data.
This is best done on a calculator!
y  ax 2  bx  c
Quadratic Regression
Cubic Regression is the process of finding a
best-fitting cubic model for a set of data.
y  ax3  bx 2  cx  d
Cubic Regression
Do your data points keep getting bigger and bigger (or
smaller and smaller)? Like exponentially bigger( or
smaller)? Try an exponential regression model.
y  a b
x
You will be able to
model direct variation
The simplest kind of line is one that has a slope
and intersects the y-axis at the origin.
• This type of line
shows direct
variation.
• Equation: y = mx
When a line shows direct variation between x
and y, we write y = kx, where k is the constant
of variation.
• y is said to vary
directly with x.
• k is the same thing
as slope!
The following are equivalent:
1. y varies directly as (or with) x
2. y is directly proportional to x
3. y = kx, for some nonzero constant k, called
the constant of variation or constant of
proportionality
The variable y varies directly as x. Write and
graph a direct variation equation that has the
given ordered pair as a solution. Then find
each constant of variation.
1. (3, -9)
2. (-7, 4)
Since y = kx can be rewritten as k = y/x, a set of
ordered pairs shows direct variation if y/x is
constant.
y = 2x
y = 2x + 1
x
1
2
3
4
y
2
4
6
8
x
1
2
3
4
y
3
5
7
9
Great white sharks have triangular teeth. The
table below gives the length of a side of a tooth
and the body length for each of six great white
sharks. Tell whether tooth length and body
length shows direct variation. If so, write an
equation that relates the quantities.
The variable y is directly proportional to the
square of x such that y = 18 when x = 6. Write
an equation that relates x and y. What is the
constant of variation? Find y when x = 12.
The variable y varies directly as the square root
of x such that y = 18 when x = 9. Write an
equation that relates x and y. What is the
constant of variation? Find y when x = 16.
The following are equivalent:
1. y varies directly as (or with) the nth power
of x
2. y is directly proportional to the nth power
of x
3. y = kxn, for some nonzero constant k
You will be able to model inverse variation
A company has found that the demand for its
product varies inversely as the price of the
product. Write an equation relating demand d
and price p. Interpret the meaning of your
model.
The following are equivalent:
1. y varies inversely as (or with) x
2. y is inversely proportional to x
k
3. y = , for some nonzero constant k
x
Your model could be inversely proportional to
the nth power of x
•
1
2
2
1
3
2/3
4
1/2
The table compares the area A (in mm2) of a computer
chip with the number c of chips that can be obtained
from a silicon wafer.
Area (mm2), A
58
62
66
70
Number of chips, c
448
424
392
376
1. Write a model that gives c as a
function of A.
2. Predict the number of chips per
wafer when the area of a chip is
81 mm2.
You will be
able to
model joint
variation
The following are equivalent:
1. z varies jointly as (or with) x and y
2. z is jointly proportional to x and y
3. z = kxy, for some nonzero constant k
Your model could be jointly proportional to the
nth power of x and the mth power of y
The maximum load that can be safely supported by
a horizontal beam is jointly proportional to the
width of the beam and the square of its depth,
and inversely proportional to the length of the
beam. Determine the change in the maximum
safe load under the following conditions:
1. The width of the beam doubles
2. The depth of the beam doubles
3. The length of the beam doubles
Find a mathematical model to represent this little know
statement from classical mechanics:
The force of gravity F between
two masses m1 and m2 is
directly proportional to the
product of the two masses
and inversely proportional to
the square of the distance
between them r.
Write a sentence using variation terminology for
the volume of a cone.
1 2
V  r h
3
Objectives:
1. To use a graphing
utility to perform a
linear regression and
use it to make a
prediction
2. Write a mathematical
model involving
direct, inverse, or
joint variation
•
•
•
•
•
•
•
Assignment:
P. 110-1: 7a, c, e;
9 a-e; 10 a-e
P. 111: 25, 31
P. 112: 39-48 S
P. 112: 49-54 S
P. 113: 55-62 S, 64, 70
P. 114: 78
HW Supplement