12.1 Regression Inference KEY POINTS
Regression Model
Unbiased Estimators for 𝜶 𝒂𝒏𝒅 𝜷.
We use 𝑦̂ = 𝑎 + 𝑏𝑥 to ________ 𝜇𝑦 = 𝛼 + 𝛽𝑥
The slope 𝒃 of the LSRL is an ____________
estimator of the true slope 𝜷.
The intercept 𝒂 of the LSRL is an unbiased
estimator of the true _____________ 𝜶.
The __________ ________ 𝒔 is an unbiased
estimator of the true standard deviation of
(𝝈𝒚 ).
Notice degrees of freedom is _____! We lose two
degrees of freedom for ______________ two
parameters (𝛼 𝑎𝑛𝑑 𝛽).
What about the standard deviations of all these normal
distributions?
The
𝜇𝑦
has
a
____________
relationship with 𝑥:
Mean: 𝜇𝑏 = 𝛽
Standard Deviation:
𝜇𝑦 = 𝛼 + 𝛽𝑥
Where: slope 𝛽 and intercept 𝛼 are _______________
parameters
For any fixed value of 𝑥, the _____________ 𝑦 varies
according to a _______________
.
Repeated responses of 𝑦 are _______________ of each
other.
The
Sampling Distribution for 𝜷
of 𝑦 (𝜎𝑦 ) is the ________ for all
values of 𝑥. (𝜎𝑦 is also an __________ parameter)
12.1 Regression Inference KEY POINTS
Hypothesis Test Statistic Value
Conditions for Regression Inference on Slope
Independent Observations
The observations are _______________.
Check that you have an ______ of data.
Linear Relationship
Notice, 𝑑𝑓 = 𝑛 − 2
The true relationship is __________.
We lose 2 degrees of freedom for ______________
two parameters!
Check the ____________ plot for ___________
scattering of points.
Hypothesis Statements for Regression Inference
Constant Standard Deviation
𝐻0 : 𝛽 = 0
The standard deviation of the _________ is constant.
This implies that there is no ________________
between x and y OR that x should not be used to
_____________ y.
Check _______________ to see if the points are
____________ spaced across LSRL.
𝐻0 : 𝛽 > 0
Responses Vary Normally
The responses vary normally about the ________
regression line.
Check the ____________ of ____________ for
symmetry and no outliers.
Confidence Interval Formula for Regression
𝐻0 : 𝛽 < 0
𝐻0 : 𝛽 ≠ 0
Where 𝜷 is the true ______ of the LSRL ... (context)
12.1 Regression Inference
EXAMPLE “Weight vs. Body Fat”
4) Explain the meaning of the slope in context of the
problem.
It is difficult to accurately determine a person’s body
fat percentage without immersing him or her in good
estimate immersed 20 male subjects, and then
measured their weights.
Find the LSRL, correlation coefficient, and coefficient
of determination.
Weight (lb)
175
181
200
159
196
192
205
173
187
188
188
240
175
168
246
160
215
159
146
219
Body Fat (%)
6
21
15
6
22
31
32
21
25
30
10
20
22
9
38
10
27
12
10
28
5) Does the y-intercept of the LSRL have meaning in
context of the problem?
6) Explain the meaning of the correlation coefficient in
context of the problem.
7) Explain the meaning of the coefficient of
determination in context of the problem.
8) Estimate 𝛼, 𝛽 𝑎𝑛𝑑 𝜎 for the problem.
9) Create a scatterplot for the data.
𝟏) 𝑳𝑺𝑹𝑳:
𝑦̂ =
𝟐) 𝑪𝒐𝒓𝒓𝒍𝒆𝒂𝒕𝒊𝒐𝒏 𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒆𝒏𝒕:
𝑟 =
10) Create a residual plot for the data.
𝟑) 𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏:
𝑟2 =
12.1 Regression Inference
EXAMPLE “Weight vs. Body Fat” (CONT.)
12) Confidence Interval Find a 95% confidence
interval for the true slope of the LSRL.
11) Significance Test Is there sufficient evidence
that weight can be used to predict body fat?
13) Computer generated data
Here is the computer generated result from the data:
𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒: 𝑛 = 20
𝑅 − 𝑠𝑞𝑢𝑎𝑟𝑒 = 43.83%
𝑠 = 7.0491323
What are the degrees of freedom?
What is the correlation coefficient?
What does the value of s represent?
What is the equation of the LSRL?
What does the value 0.060653996 represent?