Stor 155, Section 2, Last Time• Inference for Regression
– Least Square Fits
– Sampling distrib’ns for slope and intercept
• Regression Tool
– Gave many useful answers
(CIs, Hypo Tests, Graphics,…)
– But had to “translate language”
Reading In Textbook
Approximate Reading for Today’s Material:
Pages 634-667
Next Time: All review
Stat 31 Final Exam:Date & Time:
Tuesday, May 8, 8:00-11:00
Last Office Hours:• Thursday, May 3, 12:00 - 5:00• Monday, May 7, 10:00 - 5:00 • & by email appointment (earlier)
Bring with you, to exam:• Single (8.5" x 11") sheet of formulas• Front & Back OK
Prediction in Regression
Idea: Given data
Can find the Least Squares Fit Line, and do
inference for the parameters.
Given a new X value, say , what will the
new Y value be?
nn YXYX ,,,, 11
0X
Prediction in Regression
Dealing with variation in prediction:
Under the model:
A sensible guess about ,
based on the given ,
is:
(point on the fit line above )
iii ebaXY
0Y
bXaYYˆˆˆˆ 00
0X
0X
Prediction in Regression
What about variation about this guess?
Natural Approach: present an interval
(as done with Confidence Intervals)
Careful: Two Notions of this:
1. Confidence Interval for mean of
2. Prediction Interval for value of
0Y
0Y
Prediction in Regression
1. Confidence Interval for mean of :
Use:
where:
and where
0Y
YSEtY ˆ*ˆ
n
ii xx
xxn
sSEY
1
2
20
ˆ
1
)2,95.01(* nTINVt
Prediction in Regression
Interpretation of:
• Smaller for closer to
• But never 0
• Smaller for more spread out
• Larger for larger
0x x
n
ii xx
xxn
sSEY
1
2
20
ˆ
1
six
Prediction in Regression
2. Prediction Interval for value of
Use:
where:
And again
0Y
YSEtY ˆ*
0
n
ii
Y
xx
xx
nsSE
1
2
20
ˆ
11
)2,95.01(* nTINVt
Prediction in Regression
Interpretation of:
• Similar remarks to above …
• Additional “1 + ” accounts for added
variation in compared to
n
ii
Y
xx
xxn
sSE
1
2
20
ˆ
11
Y0Y
Prediction in RegressionRevisit Class Example 33,
Textbook Problem 10.23-10.25:
Engineers made measurements of the Leaning Tower of Pisa over the years 1975 – 1987. “Lean” is the difference between a points position if the tower were straight, and its actual position, in tenths of a meter, in excess of 2.9 meters. The data are listed above…
Prediction in Regression??? Next time: spruce up these examples
a lot ???
Prediction in RegressionClass Example 33,
Textbook Problems 10.23 – 10.25:
(a) Plot the data, Does the trend in lean over time appear to be linear?
(b) What is the equation of the least squares fit line?
(c) Give a 95% confidence interval for the average rate of change of the lean.
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls
Prediction in RegressionHW:
10.7 b, c, d
10.8 ((c) [11610, 12660], [9554, 14720])
Prediction in RegressionRevisit Class Example 33,
Textbook 10.23 – 10.25:
Engineers made measurements of the Leaning Tower of Pisa over the years 1975 – 1987. “Lean” is the difference between a points position if the tower were straight, and its actual position, in tenths of a meter, in excess of 2.9 meters. The data are listed above…
Prediction in RegressionClass Example 33, Problem 10.24:
(a) In 1918 the lean was 2.9071 (the coded value is 71). Using the least squares equation for the years 1975 to 1987, calculate a predicted value for the lean in 1918
(b) Although the least squares line gives an excellent fit for 1975 – 1987, this did not extend back to 1918. Why?
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls
Prediction in RegressionClass Example 33, Problem 10.25:
(a) How would you code the explanatory variable for the year 2002?
(b) The engineers working on the tower were most interested in how much it would lean if no corrective action were taken. Use the least squares equation line to predict the lean in 2005.
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls
Prediction in RegressionClass Example 33, Problem 10.25:
(c) To give a margin of error for the lean in 2005, would you use a confidence interval for the mean, or a prediction interval? Explain your choice.
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls
Prediction in RegressionClass Example 33, Problem 10.25
(d) Give the values of the 95% confidence interval for the mean, and the 95% prediction interval. How do they compare?
Recall generic formula (same for both):
SEtY *0
Prediction in Regression
Class Example 33, Problem 10.25
Difference was in form for SE:
CI for mean:
PI for value:
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg33.xls
n
ii
e
xx
xxn
sSEY
1
2
20
ˆ
1
n
ii
e
xx
xxn
sSEY
1
2
20
ˆ
11
Outliers in RegressionCaution about regression:
Outliers can have a major impacthttp://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
• Single point can throw slope way off
• And intercept too
• Can watch for this, using plot
• And residual plot show this, too
Nonlinear RegressionWhen lines don’t fit data:
• How do we know?
• What can we do?
• There is a lot…
• But beyond scope of this course
• Some indication…
Nonlinear RegressionClass Example 34: World Population
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg34.xls
Main lessons:
• Data can be non-linear
• Identify with plot
• Residuals even more powerful at this
• Look for systematic structure
Nonlinear RegressionClass Example 34: World Population
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg34.xls
When data are non-linear:
• There is non-linear regression
• But not covered here
• Can use lin. reg’n on transformed data
• Log transform often useful
Next time:Additional Issues in Regression
Robustness
Outliers via Java Applet
HW on outliers
And Now for Something Completely Different
Etymology of:
“And now for something completely
different”
Anybody heard of this before?
(really 2 questions…)
And Now for Something Completely Different
What is “etymology”?
Google responses to:
define: etymology• The history of words; the study of the history
of words.csmp.ucop.edu/crlp/resources/glossary.html
• The history of a word shown by tracing its development from another language.www.animalinfo.org/glosse.htm
And Now for Something Completely Different
What is “etymology”?
• Etymology is derived from the Greek word e/)tymon(etymon) meaning "a sense" and logo/j(logos) meaning "word." Etymology is the study of the original meaning and development of a word tracing its meaning back as far as possible.www.two-age.org/glossary.htm
And Now for Something Completely Different
Google response to: define: and now for something
completely differentAnd Now For Something Completely Different is a
film spinoff from the television comedy series Monty Python's Flying Circus. The title originated as a catchphrase in the TV show. Many Python fans feel that it excellently describes the nonsensical, non sequitur feel of the program. en.wikipedia.org/wiki/And_Now_For_Something_Completely_Different
And Now for Something Completely Different
Google Search for:
“And now for something completely different”
Gives more than 100 results….
A perhaps interesting one:
http://www.mwscomp.com/mpfc/mpfc.html
And Now for Something Completely Different
Google Search for:“Stor 155 and now for something completely different”
Gives:[PPT] Slide 1File Format: Microsoft Powerpoint - View as HTML
http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/ ... And Now for Something Completely Different. P: Dead bugs on windshield. ...stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155-07-01-30.ppt - Similar pages
Review Slippery Issues
Major Confusion:
Population Quantities
Vs.
Sample Quantities
Review Slippery Issues
Population Mathematical Notation:
(fixed & unknown)
Sample Mathematical Notation :
(summaries of data, have numbers)
p,,
psX ˆ,,
Hypothesis Testing – Z scores
E.g. Fast Food Menus:
Test
Using
P-value = P{what saw or m.c.| H0 & HA bd’ry}
(guides where to put $21k & $20k)
000,20$:0 H
000,20$: AH
10,400,2$,000,21$ nsX
Hypothesis Testing – Z scores
P-value = P{what saw or or m.c.| H0 & HA bd’ry}
rybdXP '|000,21$
000,20$|000,21$ XP
102400$
000,20$000,21$
nsX
P
317.1 ZP
Hypothesis Testing – Z scores
P-value
This is the Z-score
Computation: Class E.g. 24, Part 6http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg24.xls
Distribution: N(0,1)
317.1 ZP
Hypothesis Testing – Z scores
P-value
So instead of reporting tail probability,
Report this cutoff instead,
as “SDs away from mean $20,000”
317.1 ZP
Review for Final
An Important Mode of Thinking:
Ideas vs. “Cookbook”
Ideas vs. “Cookbook”How do you view your sheet of formulas?
• A set of recipes?
• “Look through list” to solve problems?
• Getting harder to find now?
Problems:
• Too many decisions to make
• Hard to sort out while looking through…
Ideas vs. “Cookbook”Too many decisions to make, e.g.:
• Binomial vs. Normal
• 1-sided vs. 2-sided Hypo. Tests
• TDIST (INV) vs. NORMDIST (INV)
• CI vs. Sample Size calculation
• 1 sample vs. 2 sample
• Which is H0, HA? And what direction?
• What is “m.c.”? What is “Bdry”?
Ideas vs. “Cookbook”Suggested Approach:
• Use concepts to guide choice
• This is what I try to teach
• And is what I am testing for…
How to learn?
• Go through old HW (random order)
• When stumped, look through notes
(look for main Ideas, not “the right formula”)
A useful conceptPerhaps not well taught?
|a – b| - “Number of spaces between a and b on the numberline”
e.g. Midterm II, problem 3c (x = 0, 1, 2, 3)
{|X – 1| > 1} = {“number of spaces between X and 1 is more than 1”}
= {X = 3}
A useful concepte.g. Midterm II, problem 3c (x = 0, 1, 2, 3)
{|X – 1| > 1} = {“number of spaces between X and 1 is more than 1”}
= {X = 3}
Because:
0 1 2 3
Response to a RequestYou said at the end of today's class that you would be
willing to take class time to "reteach" concepts that might still be unknown to us.
Well, in my case, it seems that probability and probability distribution is a hard concept for me to grasp.
On the first midterm, I missed … and on the second midterm, I missed …
I seem to be able to grasp the other concepts involving binomial distribution, normal distribution, t-distribution, etc fairly well, but probability is really killing me on the exams.
If you could reteach these or brush up on them I would greatly appreciate it.
A Flash from the PastTwo HW “Traps”
1. Working together:• Great, if the relationship is equal• But don’t be the “yes, I get it” person…
2. The HW “Consortium”:• You do HW 1, and I’ll do HW 2…• Easy with electronic HW• Trap: HW is about learning• You don’t learn on your off weeks…
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