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Latin
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Latin/Greek Root Words
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(Statistics
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Chapter 14: Regression Significance Testing
AP Statistics Standards
V. Statistical Inference: Estimating population parameters
and testing hypotheses (continued)
B. Tests of significance
7.
Test for the slope of a least-squares regression line
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| Essential Question:
How can you express the
uncertainty in the slope of a line? |
Ch. 14.1 Inference for Regression
Relevance:
Regression analysis (generally multiple regression analysis) is
a very common form of data analysis found in technical journals--often
the primary source of information for research papers. It is not
possible to read and understand them without an understanding of
inference for regression..
- State the 2
inferences drawn when using regression data.
(slope & intercept)
- State the assumptions for
regression inference.
- for any x value, y-data is normally distributed
(remember, Xs are perfect Ys are not.)
- for any x value, the y-data's standard deviation is the same
- the means of the y-data distribution at any value of x form a
straight line relationship: my =
a + bx
- Calculate the standard error of the least squares regression
line.
s = [
S(y - ŷ)2
/ (n-2) ]1/2
- Calculate the standard error of the slope.
SEb = s / [ (S(x -
xbar)2 ]1/2
Homefun -- Read section 14.1, prob. 14.5, 14.7, 14.9
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- Lesson 1
- Key Concept: How to
evaluate the slope obtained in a regression
analysis
Interactive
Discussion: Why is the slope of the line is a big deal. It often
has physical meaning
Famous slopes Mr. R
has known:
2-person teams: Using
data from an Einstein's photoelectric experiment find plank's
constant using regression analysis in Minitab. Calculate a
confidence interval on the slope of the line (plank's constant).
Compare this technique to IB error analysis techniques.
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| Essential Question:
How can you spot a meaningless
regression anaysis? |
Evaluating
Regression Results for the Slope
- Generate a confidence interval for the slope.
(n-2 degrees of
freedom)
b ± t*SEb
- Calculate the t-value for a hypothesis test
of the slope Ho: b
= 0. (n-2 degrees of freedom)
t = b / SEb
- Perform a significance test for the slope of a least squares regression line.
Ho: β = 0
Ha: β ≠ 0, β < 0,
β > 0
Note: most
computer programs return a p-value for Ha: β ≠ 0. For β < 0
or
β > 0, simply cut the p-value in half
- Perform a significance test for the
intercept of a least squares regression line.
- Ho:
a = 0
Note: often
a
does equal 0. A high
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p-value indicates that the intercept
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is essentially 0.
Ha:
a ≠ 0,
a < 0,
a > 0
Note: most
computer programs return a p-value for Ha:
a ≠ 0.
For
a < 0
or
a > 0,
simply cut the p-value in half
- Correctly perform least squares regression
using Minitab.
- Correctly interpret least squares regression
computer output (such as from Minitab).
Homefun --
prob. 14.11, 14.17 |
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- Lesson 2
- Key Concept: Spotting a
meaningless regression analysis
Interactive
Discussion: Objectives. While it's possible to spot a
meaningless regression analysis, it's not possible to tell for
certain if an analysis is meaningful.
2-person teams: (see the stats investigation below)
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| Essential Question:
How can you account for
variability in a predicted data point from a regression analysis? |
Evaluating
Regression Results for a Given y-hat
- Generate a
confidence interval for the
average y-hat.
(n-2 degrees of freedom, see page 797)
y-hat ± t*SEμ-hat
- Generate a
prediction interval for
a y-hat at a specific value of x.
(n-2 degrees of freedom, see page 797)
y-hat ± t*SEy-hat
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| Stats
Investigation: How to Spot a
Meaningless Regression Analysis - time approx 2 class
periods (individual work) |
Purpose: Determine if a regression analysis using random
numbers that has a high r-square value can be detected with
hypothesis tests on the slope and intercept.
Instructions: Remember the
stats investigation you did earlier in which you determined
that even random data can produce a high r-square value.
Redo the regression/correlation analysis in Minitab on the 4
sets of data you saved and interpret the results. Be
sure to take all the recommended steps for producing a
statistically significant regression analysis.
Questions /Conclusions:
- Based on your data, could you spot
randomness with the hypothesis tests on the slope and
intercept of the regression equation.
Explain
- Outline all the steps which should be taken to produce
a regression/correlation analysis with the best chance of
being meaningful.
- Can a thorough statistical analysis of
bivariate data, by itself
fully establish that a regression result is meaningful?
Explain
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| Essential Question:
How can we evaluate or include
possible lurking variable in a
regression anaysis? |
Ch. 14
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Perform multiple linear
regression analysis using Minitab and correctly interpret the
regression equation, R2, and the hypothesis tests.
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State how you would plot and interpret
residuals for a multiple regression analysis.
Remember, a residual = (yi - yhat).
Although there are several x-variables each with its own value, there
is only one y-value and only one yhat. Hence, the residual is normally
plotted against the y-value.
Classfun:
Chapter 8, AMSCO book Review Exercise 1-6 p. 254, 1-7 p. 257, 1-6 259,
1-5 262
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- Lesson 3
- Key Concept: Control of variables in a study
Interactive Discussion:
Review the following from
chap 5:
- Be as one with the three basic principles of
experimental design.
- Control - effects of lurking
variables
- Randomization - prevents sampling
bias
- Replication - collect numerous data
points
2-person teams: Perform a multiple
linear regression analysis on the SAT data used earlier in the year.
Compare the results with those obtained earlier using 2 variable
regression analysis.
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