Transcript Finding Areas with Calc 1. Shade Norm

Describing Bivariate
Relationships
Chapter 3 Summary
YMS
AP Stats
3.1 Response Vs.
Explanatory Variables
•
Response variable measures an outcome of a study,
explanatory variable helps explain or influences
changes in a response variable (like independent vs.
dependent).
•
Calling one variable explanatory and the other
response doesn’t necessarily mean that changes in
one CAUSE changes in the other.
•
Ex: Alcohol and Body temp: One effect of Alcohol is a
drop in body temp. To test this, researches give
several amounts of alcohol to mice and measure each
mouse’s body temp change. What are the
explanatory and response variables?
Scatterplots
•
Scatterplot shows the relationship between two quantitative variables measured
on the same individuals.
•
Explanatory variables along X axis, Response variables along Y.
•
Each individual in data appears as the point in the plot fixed by the values of
both variables for that individual.
•
Example:
Examining Scatterplots
Overall pattern
•
Direction
•
Form
•
Strength
•
Outliers or deviations
Interpreting Scatterplots
•
Direction: in previous example, the overall pattern moves
from upper left to lower right. We call this a negative
association.
•
Form: The form is slightly curved and there are two
distinct clusters. What explains the clusters? (ACT
States)
•
Strength: The strength is determined by how closely the
points follow a clear form. The example is only
moderately strong.
•
Outliers: Do we see any deviations from the pattern?
(Yes, West Virginia, where 20% of HS seniors take the
SAT but the mean math score is only 511).
Association
Introducing Categorical
Variables
Calculator Scatterplot
moth
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
DegreeDay
24
51
43
33
26
13
4
0
0
1
6
12
30
32
52
30
Gas
(100cuft)
6.3 10.9 8.9 7.5 5.3 4.0 1.7 1.2 1.2 1.2 2.1 3.1 6.4 7.2 11.0 6.9
•
Enter the Degree-Days in L1 and Gas in L2
•
Next specify scatterplot in Statplot menu (first graph). X list
L1 Y List L2 (explanatory and response)
•
Use ZoomStat.
•
Notice that their are no scales on the axes and they aren’t
labeled. If you are copying your graph to your paper,
make sure you scale and label the Axis (use Trace)
Correlation
•
r
The Correlation measures the direction and strength of the linear relationship
between 2 variables.
Z Z
x
y
•
Formula- (don’t need to memorize or use): r =
•
In Calc: Go to Catalog (2nd, zero button), go to DiagnosticOn, enter, enter. You
only have to do this ONCE! Once this is done:
•
Enter data in L1 and L2 (you can do calc-2 var stats if you want the mean and sd
of each)
•
Calc, LinReg (A + Bx) enter
n 1
Interpreting Correlation
•
Caution- our eyes can be fooled! Our eyes are not
good judges of how strong a linear relationship is.
The 2 scatterplots depict the same data but drawn
with a different scale. Because of this we need a
numerical measure to supplement the graph.
Interpreting r
•
The absolute value of r tells you the strength of the
association (0 means no association, 1 is a strong
association)
•
The sign tells you whether it’s a positive or a negative
association. So r ranges from -1 to +1
•
Note- it makes no difference which variable you call x and which you
call y when calculating correlation, but stay consistent!
•
Because r uses standardized values of the observations, r does not
change when we change the units of measurement of x, y, or both.
(Ex: Measuring height in inches vs. ft. won’t change correlation with
weight)
•
values of -1 and +1 occur ONLY in the case of a perfect linear
relationship , when the variables lie exactly along a straight line.
Examples
1. Correlation requires that both variables be
quantitative
2. Correlation measures the strength of only
LINEAR relationships, not curved...no matter
how strong they are!
3. Like the mean and standard deviation, the
correlation is not resistant: r is strongly affected
by a few outlying observations. Use r with
caution when outliers appear in the scatterplot
4. Correlation is not a complete summary of
two-variable data, even when the relationship is
linear- always give the means and standard
deviations of both x and y along with the
correlation.
3.3- least squares
regression
Text
The slope here B = .00344 tells us that fat
gained goes down by .00344 kg for each
added calorie of NEA according to this
linear model. Our regression equation is
the predicted RATE OF CHANGE in the
response y as the explanatory variable x
changes.
The Y intercept a = 3.505kg is the fat gain
estimated by this model if NEA does not
change when a person overeats.
Prediction
•
We can use a regression line to predict the response y
for a specific value of the explanatory variable x.
LSRL
•
In most cases, no line will pass exactly through all
the points in a scatter plot and different people will
draw different regression lines by eye.
•
Because we use the line to predict y from x, the
prediction errors we make are errors in y, the
vertical direction in the scatter plot
•
A good regression line makes the vertical distances of the
points from the line as small as possible
•
Error: Observed response - predicted response
LSRL Cont.
Equation of LSRL
•
Example: The Sanchez household is about to install solar
panels to reduce the cost of heating their house. In order to
know how much the panels help, they record their
consumption of natural gas before the panels are installed.
Gas consumption is higher in cold weather, so the
relationship between outside temp and gas consumption is
important.
Facts about LeastSquares regression
•
The distinction between explanatory and response variables is essential
in regression. If we reverse the roles, we get a different least-squares
regression line.
•
There is a close connection between corelation and the slope of the
LSRL. Slope is r times Sy/Sx. This says that a change of one standard
deviation in x corresponds to a change of 4 standard deviations in y.
When the variables are perfectly correlated (4 = +/- 1), the change in the
predicted response y hat is the same (in standard deviation units) as the
change in x.
•
The LSRL will always pass through the point (X bar, Y Bar)
•
r squared is the fraction of variation in values of y explained by the x
variable
•
Describe the direction, form, and strength of the relationship
•
Positive, linear, and very strong
•
About how much gas does the regression line predict that the family will
use in a month that averages 20 degree-days per day?
•
•
500 cubic feet per day
How well does the least-squares line fit the data?
R squared- Coefficient
of determination
If all the points fall directly on the least-squares
line, r squared = 1. Then all the variation in y
is explained by the linear relationship with x.
So, if r squared = .606, that means that 61% of
the variation in y among individual subjects is
due to the influence of the other variable. The
other 39% is “not explained”.
r squared is a measure of how successful the
regression was in explaining the response
3.3 Influences
•
Correlation r is not resistant. Extrapolation is not
very reliable. One unusual point in the scatterplot
greatly affects the value of r. LSRL also not
resistant.
•
A point extreme in the x direction with no other
points near it pulls the line toward itself. This point
is influential.
Lurking VariablesBeware!
•
Example: A college board study of HS grads found a strong correlation
between math minority students took in high school and their later
success in college. News articles quoted the College Board saying that
“math is the gatekeeper for success in college”.
•
But, Minority students from middle-class homes with educated
parents no doubt take more high school math courses. They are also
more likely to have a stable family, parents who emphasize
education, and can pay for college etc. These students would likely
succeed in college even if they took fewer math courses. The family
background of students is a lurking variable that probably explains
much of the relationship between math courses and college success.
Residuals
•
The error of our predictions, or vertical distance
from predicted Y to observed Y, are called
residuals because they are “left-over” variation in
the response.
One subject’s NEA rose by 135 calories.
That subject gained 2.7 KG of fat. The
predicted gain for 135 calories is
Y hat = 3.505- .00344(135) = 3.04 kg
The residual for this subject is
y - yhat
= 2.7 - 3.04 = -.34 kg
Residual Plot
•
The sum of the least-squares residuals is always zero.
•
The mean of the residuals is always zero, the horizontal line
at zero in the figure helps orient us. This “residual = 0” line
corresponds to the regression line
Examining Residual Plot
•
Residual plot should show no obvious pattern. A curved
pattern shows that the relationship is not linear and a straight
line may not be the best model.
•
Residuals should be relatively small in size. A regression line
in a model that fits the data well should come close” to most
of the points.
•
A commonly used measure of this is the standard deviation
of the residuals, given by:
s
For the NEA and fat gain data, S =
7.663
 .740
14
 residuals
n2
2
Residuals List on Calc
•
If you want to get all your residuals listed in L3
highlight L3 (the name of the list, on the top)
and go to 2nd- stat- RESID then hit enter and
enter and the list that pops out is your resid for
each individual in the corresponding L1 and
L2. (if you were to create a normal scatter plot
using this list as your y list, so x list: L1 and Y list
L3 you would get the exact same thing as if you
did a residual plot defining x list as L1 and Y list
as RESID as we had been doing).
This is a helpful list to have to check your work when
asked to calculate an individuals residual.
Residual Plot on Calc
•
Produce Scatterplot and Regression line from data (lets use
BAC if still in there)
•
Turn all plots off
•
Create new scatterplot with X list as your explanatory
variable and Y list as residuals (2nd stat, resid)
•
Zoom Stat
Bivariate Relationships
What is Bivariate data?
When exploring/describing a bivariate (x,y) relationship:
Determine the Explanatory and Response variables
Plot the data in a scatterplot
Note the Strength, Direction, and Form
Note the mean and standard deviation of x and the
mean and standard deviation of y
Calculate and Interpret the Correlation, r
Calculate and Interpret the Least Squares
Regression Line in context.
Assess the appropriateness of the LSRL by
constructing a Residual Plot.