
Latin
Latin/Greek Root Words


(Statistics
connection) 

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 leastsquares regression line


Essential Question:
Why is finding the slope
of a straight line a big deal? 
The Awesomeness of Slope
Famous slopes Mr. R
has known:

electrical
conductivity
 COF
 spring constant

Plank's constant
 density
 g
 the perfect gas law
constant
 etc. etc. etc

Formative Assessment: 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.

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 journalsoften
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 (b) & intercept
(a),
these are both parameters, hence, the Greek letters.
 State the assumptions for
regression inference.
 for any x value, ydata is normally distributed.
Remember, Xs are perfect Ys
are not.
 for any x value, the ydata's standard deviation is the same.
See objective 3 below.
 the means of the ydata distribution at any value of x form a
straight line relationship:
m_{y} =
a + bx
 Calculate the standard error of the least squares regression
line.
s = [
S(y  ŷ)^{2
}/ (n2) ]^{1/2}
 Calculate the standard error of the slope.
SE_{b} = s / [ (S(x 
xbar)^{2 }]^{1/2}
Homefun  Read section 12.1 Exercise 1, 3, 7, 15 pp. 759 to 762


Essential Question:
How can you spot a meaningless
regression anaysis? 
Evaluating
Regression Results for the Slope
 Generate a confidence interval for the slope.
(n2 degrees of
freedom) Note, b is the estimate of slope from performing regression
analysis.
β
is a parameter and is the actual slope.
b
± t^{*}SE_{b}
 Calculate the tvalue for a hypothesis test
of the slope H_{o}: b
= 0.
(n2 degrees of freedom)
t = (
b 
0 ) / SE_{b
or }t =
b / SE_{b}
 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 pvalue for Ha: β ≠ 0 (see
drawing at right).
For β < 0 or β
> 0, simply cut the pvalue 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

pvalue indicates that the intercept

is essentially 0. It does not
 necessarily
mean that the intercept is meaningless.
Ha:
a ≠ 0,
a < 0,
a > 0
Note: most
computer programs return a pvalue for Ha:
a ≠ 0.
For
a < 0
or
a > 0,
simply cut the pvalue in half
 Correctly perform least squares regression
using Minitab.
 Correctly interpret least squares regression
computer output (such as from Minitab).
Homefun  Read section 12.2 Exercise
33, 35,37 pp. 786 to 788


Essential Question:
How can you account for
variability in a predicted data point from a regression analysis? 
Evaluating
Regression Results for a Given yhat
 Generate a
confidence interval for the
average yhat give from
averaging numerous yvalues
obtained at a specific value of x.
(n2 degrees of freedom)
yhat ± t^{*}SE_{μhat}
SE_{μhat} = s [ 1/n + ( x*  xbar )^2 / (S(x 
xbar)^{2}) ]^0.5
 Generate a
prediction interval
for a yhat
from
a single y value
obtained at a specific value of x.
(n2 degrees of freedom)
yhat ± t^{*}SE_{yhat}
SE_{yhat} = s [ 1 + 1/n + ( x*  xbar )^2 / (S(x 
xbar)^{2}) ]^0.5


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 rsquare 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 rsquare 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



Essential Question:
How can we evaluate or include
possible lurking variable in a
regression anaysis? 
Ch. 14

Perform multiple linear
regression analysis using Minitab and correctly interpret the
regression equation, R^{2}, and the hypothesis tests.

State how you would plot and interpret
residuals for a multiple regression analysis.
Remember, a residual = (y_{i } yhat).
Although there are several xvariables each with its own value, there
is only one yvalue and only one yhat. Hence, the residual is normally
plotted against the yvalue.
Classfun:
Chapter 8, AMSCO book Review Exercise 16 p. 254, 17 p. 257, 16 259,
15 262
Homefun  Chapter 12 practice test (all parts) pp. 796 to 798
Cumulative AP Practice Test pp. 799 to 806


