Download Chapter 3

Bivariate Data and
Scatter Plots
Bivariate Data: The values of two different variables that are
obtained from the same population element.
While the variables may be either categorical or quantitative,
we will focus on cases where they are both quantitative.
Can we predict values of one variable from values of the
other variable?
Do the values of one variable cause the values of the other
variable?
Section 3.1, Page 59
1
Scatter Plot Example
TI-83
Scatter Plots always have
and explanatory variable and
a response variable. The
choice is arbitrary. The
explanatory variable is
always plotted on the x-axis,
and the response variable is
always plotted on the y axis.
STAT – EDIT – ENTER; Enter x data in L1, and y in L2
2nd STAT PLOT – ENTER -1: Plot 1
Highlight ON
Type: Highlight first icon
XList: 2nd L1
YList 2nd L2
ZOOM 9: ZoomStat
TRACE; Use arrows to move to points and display values.
Section 3.1, Page 60
2
Linear Correlation
Linear Correlation: A measure of the strength of a linear
relationship between two variables. The closer to a
straight line the dots are, the stronger the relationship.
If there correlation, then we say the two variables
are associated. Changes in the value of one
variable are associated with changes in the value
of the other variable.
Section 3.1, Page 61
3
Coefficient of Correlation
Measure of Strength
Zx Zy
(x  x )
(y  y )
r
where Z x 
; Zy 
n 1
sx
sy
1  r 1;
r  1 perfect straight line negative slope
no relationship at all
r0
r 1
perfect straight line with positive slope
Also known as the Pearson Correlation
Coefficient.
Section 3.2, Page 62
4
Problems
Problems, Page 71
5
Correlation Coefficient
TI-83 Add-In Program
Finding r.
STAT – EDIT – ENTER: Enter data in L1 and L2
PRGM-CORRELTN
2nd LI – Comma – 2nd L2
SCATTER PLOT? – 1=YES; (Displays scatter plot)
ENTER; (Displays: r=.8394)
This is a moderately strong positive relationship.
Section 3.2, Page 62
6
Association and Causality
Elementary School Students
Reading Scores
8
Grade
Level
4
1
1
4
Shoe Size
8
Is this a reasonable association?
Does giving students bigger shoes cause reading
scores to improve?
What explains this association?
Lurking Variable: A third variable, often
unexpressed, that has an effect on either or both
x and y variables making it appear they are
related.
Association alone can never establish causality!
Section 3.2, Page 63
7
Problems
Problems, Page 71
8
Problems
Problems, Page 72
9
Problems
Problems, Page 72
10
Linear Regression
Line of Best Fit
If a straight line model seems appropriate, the
best fit straight line is found by using the method
of least squares. Suppose that yˆ  a  bx is the
equation of a straight line, where yˆ (read “y-hat)
represents the predicted value of y that
corresponds to a particular value of x. The least
that we find the
squares criteria requires
2
constants, a and b suchthat (y  yˆ ) is as small
as possible.

yˆ  a  bx

Section 3.3, Page 65
11
Line of Best Fit
The best line will be the one where the sum of the squares of the
“misses” is at a minimum. Calculus procedures are used to find the
coefficients, a and b such that the line ŷ = a + bx has the least squares.
br
sy
sx
r is the correlation coefficient, sy is the standard
deviation of y-values and sx is the standard
deviation of the x values

Section 3.3, Page 66
12
Linear Regression
TI-83 Add-In Program
a. For the above data, make a scatter plot, and
comment on the suitability of the data for
regression analysis.
STAT – EDIT; Enter Height in L1, and Weight in L2.
PRGN – REGBASIC
X LIST=2ND L1; Y LIST=2ND L2
SCATTER PLOT: 1=YES
The pattern looks positive,
linear, and no outliers which
could cause problems.
Scatter Plot
Section 3.3, Page 68
13
Linear Regression
TI-83 Add-In Program
b. Find the regression equation and r.
ENTER; The program is paused to view graph, hitting
ENTER moves the program along.
The equation is:
yˆ =-186.4706 + 4.7059x
r, the coefficient of
correlation = .7979, a
relatively strong relationship.
c. Check the plot of the regression line versus the
scatter plot.
ENTER – 1=YES
Section 3.3, Page 68
14
Linear Regression
TI-83 Add-In Program
d. What is the value of the slope of the line, and what
does it mean?
b = 4.7095 is the slope of the line. It indicates the
number of units change in the y value for every one
unit increase in the x value. In this problem, for each
one inch increase in height, weight increases by
4.7095 lbs. Its units are lbs/inch.
e. What is the value of the intercept of the line, and what
does it mean?
a = -186.4706 is the y intercept. It has no meaning in
this problem. It would be the weight of a person of
zero height.
f. What is the value of r2?
It is called the index of determination. It measures the
strength of the model, 1 being perfect and 0 being
useless.
r2 = .6367 indicating a relative strong positive
correlation.
Section 3.3, Page 68
15
Linear Regression
TI-83 Add-In Program
g.
Check the residual plot and explain what it means
ENTER; 1 = YES
The horizontal line represents
the regression line. For each
actual value of x, the residual is
the actual y-value – predicted
y-value. The dots show the
“misses” or residuals.
If the residuals show some kind
of a pattern, it means that the
linear regression model is not
appropriate for the data, so
other model, i.e. quadratic, may
be better. Since there is not
pattern is this plot, the linear
model is appropriate for this
data.
Section 3.3, Page 68
16
Linear Regression
TI-83 Add-In Program
h. Use the model to predict the weight of a woman who
is 65 inches tall.
PREDICTED Y: 1 = YES
X=65
Answer: 119.4 lbs
i.
Use the model to predict the weight of a woman who
is 77 inches tall.
ENTER: 1 = YES
X=77
Answer 175.9 lbs.
Notice that the range of the
x values is from 61 to 69
inches. 77 inches is too far
above the actual values
used to develop the model.
While the result is
mathematically correct, the
result is not valid in the
context of the problem.
Section 3.3, Page 68
17
Problems
Problems, Page 72
18
Problems
a.
b.
c.
d.
e.
f.
g.
h.
i.
Construct a scatter diagram.
Does the pattern appear linear?
Find the equation of best fit.
What is the value of r and what does it mean?
What is the slope? What are its units? Interpret
its meaning.
What is the y-intercept value? What does it
mean?
What does the residual plot show? What does it
mean?
Estimate the the stride rate for a speed of 19.2
ft/sec. Is the estimate reliable? Why?
Estimate the stride rate for a speed of 31 ft/sec.
Is the estimate reliable? Why?
Problems, Page 73
19
Problems
c.
d.
e.
f.
g.
h.
What is the value of r and what does it mean?
What is the slope? What are its units? Interpret
its meaning.
What is the y-intercept value? What does it
mean?
What does the residual plot show? What does it
mean?
Estimate the # of intersections for a state with
450 miles. Is the estimate reliable? Why?
Estimate the # of intersections for a state with
950 miles. Is the estimate reliable? Why?
Problems, Page 73
20
Problems
a.
b.
c.
d.
e.
f.
g.
h.
Construct a scatter diagram. What does it
indicate to you?
Find the equation of best fit.
What is the value of r and what does it mean?
What is the slope? What are its units? Interpret
its meaning.
What is the y-intercept value? What does it
mean?
What does the residual plot show? What does it
mean?
Estimate the price of an 8 year old car. Is the
estimate reliable? Why?
Estimate price of a 22 year old car. Is the
estimate reliable? Why?
Problems, Page 73
21