Download 11.1 Variation in the Slope from Sample to Sample

Chapter 11
Inference for
Regression
Section 11.1
Variation in the Slope
from Sample to Sample
Using Samples
Anytime we take a sample from the
population, our result is some type of an
_________.
Using Samples
Anytime we take a sample from the
population, our result is some type of an
estimate.
Using Samples
Anytime we take a sample from the
population, our result is some type of an
estimate.
To obtain exact answers, we would have to
take a _______ of the entire population.
Using Samples
Anytime we take a sample from the
population, our result is some type of an
estimate.
To obtain exact answers, we would have to
take a census of the entire population.
Consider the scatterplots below.
In which scatterplots is it reasonable to
model the relationship between y and x with a
line?
Consider the scatterplots below.
In which scatterplots is it reasonable to
model the relationship between y and x with a
line?
I, III, IV, V (must be linear pattern)
Consider the scatterplots below.
If you fit a line through each scatterplot by
the method of least squares, which plot will
give a line with slope closest to 0?
Consider the scatterplots below.
If you fit a line through each scatterplot by
the method of least squares, which plot will
give a line with slope closest to 0?
IV (points most scattered)
Consider the scatterplots below.
In which scatterplots is it meaningful to use
correlation to describe the relationship
between y and x?
Consider the scatterplots below.
In which scatterplots is it meaningful to use
correlation to describe the relationship
between y and x?
I, III, IV, V (must be linear pattern)
Consider the scatterplots below.
Which plot shows a correlation coefficient
closest to 1?
Consider the scatterplots below.
Which plot shows a correlation coefficient
closest to 1?
V (points packed tightest to LSRL)
To summarize a linear relationship, we
used a least squares regression line
(LSRL).
y = b0 + b1x
To summarize a linear relationship, we used
a least squares regression line (LSRL).
y = b0 + b1x
For this model to be complete, what do you
have to have?
To summarize a linear relationship, we used
a least squares regression line (LSRL).
y = b0 + b1x
For this model to be complete, what do you
have to have? The complete population
To summarize a linear relationship, we used
a least squares regression line (LSRL).
y = b0 + b1x
If you only have a random sample, the
values of b0 and b1 are estimates of true
population parameters.
“True” linear relationship is:
response = prediction from true regression
line + random deviation
“True” linear relationship is:
response = prediction from true regression
line + random deviation
Need random deviation because the point
may not fall exactly on the true regression
line
- some will be above, some below, and
some on the line
“True” linear relationship is:
response = prediction from true regression
line + random deviation
y = ( 0   1x)  
“True” linear relationship is:
response = prediction from true regression
line + random deviation
y = ( 0   1x)  
where  0 and  1 refer to the intercept and
slope of the true regression line if you had
data for the entire population.
“True” linear relationship is:
response = prediction from true regression
line + random deviation
y = ( 0   1x)  
where  0 and  1 refer to the intercept and
slope of the true regression line if you had
data for the entire population.
 is size of random deviation – how far point
falls above or below true regression line
True regression line aka line of means aka
line of averages is written:
True regression line aka line of means aka
line of averages is written:
y = ( 0   1x)  
Suppose we wanted to study the
relationship of children’s heights vs their
ages in years.
Suppose we wanted to study the
relationship of children’s heights vs their
ages in years.
Which variable goes on which axis?
Horizontal axis:
Vertical axis:
Suppose we wanted to study the
relationship of children’s heights vs their
ages in years.
Which variable goes on which axis?
Horizontal axis: ages in years
Vertical axis: children’s heights
Suppose we wanted to study the
relationship of children’s heights vs their
ages in years.
On average, kids from ages 8 to 13 grow
taller at the rate of 2 inches per year.
Heights of 8-year-olds average about 51
inches. At each age, the heights are
approximately normal, with a standard
deviation of roughly 2.1 inches.
On average, kids from ages 8 to 13 grow taller at the rate
of 2 inches per year. Heights of 8-year-olds average about
51 inches. At each age, the heights are approximately
normal, with a standard deviation of roughly 2.1 inches.
Average Height
Age, x
from Model, µy
8
?
9
10
11
12
On average, kids from ages 8 to 13 grow taller at the rate
of 2 inches per year. Heights of 8-year-olds average about
51 inches. At each age, the heights are approximately
normal, with a standard deviation of roughly 2.1 inches.
Average Height
Age, x
from Model, µy
8
51
9
10
11
12
On average, kids from ages 8 to 13 grow taller at the rate
of 2 inches per year. Heights of 8-year-olds average about
51 inches. At each age, the heights are approximately
normal, with a standard deviation of roughly 2.1 inches.
Average Height
Age, x
from Model, µy
8
51
9
53
10
11
12
On average, kids from ages 8 to 13 grow taller at the rate
of 2 inches per year. Heights of 8-year-olds average about
51 inches. At each age, the heights are approximately
normal, with a standard deviation of roughly 2.1 inches.
Average Height
Age, x
from Model, µy
8
51
9
53
10
55
11
57
12
59
If we collected a random sample of the
heights of children, we would have
different heights for each year of age.
Conditional distribution of y given x
refers to all the values of y for a fixed
value of x
Conditional distribution of y given x
refers to all the values of y for a fixed
value of x
Each conditional distribution of height for a
given age has:
Mean: y for a population and y for sample
Each conditional distribution of height for a
given age has:
Mean: y for a population and y for sample
Measure of variability:  for a population
and s for sample
Linear model is appropriate for set of data if:
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
2) Variability is about the same for each
conditional distribution
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
2) Variability is about the same for each
conditional distribution
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
2) Variability is about the same for each
conditional distribution
Page 744, D4
Linear model is appropriate for set of data if:
1) Conditional means fall near a line
2) Variability is about the same for each
conditional distribution
Page 744, D4
Two potential problems here.
First, the data appear to have curvature. The
centers of the conditional distributions of y
do not lie on a straight line, which is one of
the conditions for a linear fit.
Page 744, D4
Second, the conditional distribution of
responses at x = 2 has far greater variation
than either of the other two conditional
distributions.
The assumption of equal variances of
responses across all values of x is violated.
Common variability of y at each x is called
.
Common variability of y at each x is called
.
It is estimated by s, which can be thought of
as the standard deviation of the residuals.
Common variability of y at each x is called
.
It is estimated by s, which can be thought of
as the standard deviation of the residuals.
Spread in values of x is measured by:
Recall, most of the time the theoretical
slope, β1, is _________.
Recall, most of the time the theoretical
slope, β1, is unknown.
Recall, most of the time the theoretical
slope, β1, is unknown.
So, we use ___ to estimate β1.
Recall, most of the time the theoretical
slope, β1, is unknown.
So, we use b1 to estimate β1.
The slope b1 _____ from sample to sample.
The slope b1 varies from sample to sample.
Is this variation a good thing or not so good?
The slope b1 varies from sample to sample.
Is this variation a good thing or not so good?
Not so good as the variation affects our
predictions.
The slope b1 varies from sample to sample.
Bold line is true regression line
for the population.
Page 741, D2
Page 741, D2
The parameter,  1 , is the slope of the
theoretical model and generally will not be
known.
Page 741, D2
The statistic, b1, is an estimate of  1 that is
calculated from observed data.
The value of b1 will vary from sample to
sample.
Page 741, D2
Further, the theoretical model takes into
account that not every point is expected to
lie on the line. An error term is built in.
Epsilon, , is the deviation of a given point
from the true regression line.
Page 741, D2
Further, the theoretical model takes into
account that not every point is expected to
lie on the line. An error term is built in.
Epsilon, , is the deviation of a given point
from the true regression line.
Whereas
is the residual from the
regression line calculated from the sample
data.
Page 749, P1
Page 749, P1
Fat contains 9 calories per gram. The
theoretical slope, β1, would be 9 calories
per gram of fat.
Page 749, P1
What is the slope, b1, of the regression line?
Page 749, P1
What is the slope, b1, of the regression line?
b1 ≈ 14.9 calories per gram of fat.
Page 749, P1
b. Intercept of line tells us the number of
calories associated with a serving
containing no grams of fat.
What is the intercept, b0, of the regression
line?
Page 749, P1
b. Intercept of line tells us the number of
calories associated with a serving
containing no grams of fat.
What is the intercept, b0, of the regression
line?
b0 ≈ 111.6
Page 749, P1
c. Pizza has calories from carbohydrates
and protein as well as fat. There may be
other sources of variation in the
measurement process.
Page 749, P2
Page 749, P2
mean heightarm span = arm span
Page 749, P2
b. LSRL: Height = 7.915 + 0.952 Arm Span
Page 749, P2
b. LSRL: Height = 7.915 + 0.952 Arm Span
The interpretation of the estimated slope is
that height tends to increase by about 0.952
cm for each 1-cm increase in arm span.
Page 749, P2
b. LSRL: Height = 7.915 + 0.952 Arm Span
The estimated slope, b1, of 0.952 is very
close to Leonardo’s theoretical slope, β1 of
1.
Page 749, P2
b. LSRL: Height = 7.915 + 0.952 Arm Span
The estimated slope of 0.952 is very close to
Leonardo’s theoretical slope of 1.
The estimated intercept, b0, is 7.915 cm,
which is quite far from Leonardo’s
theoretical intercept, β0, of 0.
Page 749, P2 (c)
Mean height arm span = 0 + arm span +
Page 749, P2 (c)
Mean height arm span = 0 + arm span +
So,
= mean height arm span – arm span
Page 749, P2 (c)
Mean height = 0 + arm span +
So,
= mean height – arm span
Use L3 = L2 – L1
How do you plot the random deviations?
STAT PLOTS
1: Plot 1 ….On
Type: select scatterplot
Xlist: L1
Ylist: L3
Mark:
Page 749, P2 (c)
Page 749, P2 (c)
This plot shows the random deviations
from the theoretical line y = x plotted against
the x-values. The plot shows a pattern: All of
the negative deviations are for larger arm
spans, and the plot shows a downward
trend.
Page 749, P2 (d)
How do you calculate the residuals?
Page 749, P2 (d)
How do you calculate the residuals?
8: LinReg(a + bx) L1, L2
Page 749, P2 (d)
How do you plot the residuals?
STAT PLOTS
1: Plot 1 ….On
Type: select scatterplot
Xlist: L1
Ylist: RESID
Mark:
Page 749, P2 (d)
How do you plot the residuals?
STAT PLOTS
1: Plot 1 ….On
Type: scatterplot
Xlist: L1
Ylist: RESID [2nd, LIST, NAMES, 7:RESID]
Mark:
Page 749, P2
d.
Page 749, P2
Page 749, P3
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Page 749, P3
Page 749, P3
Page 749, P3
Questions?