Stat 155, Section 2, Last Time • Normal Distribution: • Interpretation: 68%-95%-99.7% rule • Computation of areas (frequencies) • Inverse Normal area computation • Diagnostics (for Normal approximation) • Normal Quantile plot (linear?) • Relations between variables • Scatterplots – useful visualization
Reading In Textbook Approximate Reading for Today’s Material: Pages 102-112, 123-127, 132-145 Approximate Reading for Next Class: Pages 151-163, 173-179, 192-196
E-mail Special Request Date: Mon, 29 Jan 2007 19:37:30 -0500 Subject: special request Professor Marron, I was wondering if you could do a problem like C5 in class tomorrow? I went through the notes, and I went through the workbook for Excel but it seems as though I just can't seem to get how to make a normal standard distribution. Thank you.
An Earlier Email Date: Thu, 25 Jan 2007 08:34:34 -0500 (EST) > I don't understand how you draw a density curve on excel without having data > points. In the NORMDIST function, you need data to go along with u, s, and > False. How do you draw it without, or where is the data? Right, in the class example that we considered (Stor155Eg8Done.xls), we thought about fitting a normal curve to a data set. We did this by taking the mean, and s.d. of the data set, and then using that to generate the appropriate memmber of the family of normal curves. Problem C5 is in some sense easier, since bascically the work fo calculationg the mean and s.d. is already done (note they are given as 63.1 and 4.8). So you only need to go through the other steps of generating the graphics input. I guess that one question that will come up is "what to use for endpoints of the x grid?" You could experiment a bit, but usually mean +- 3 s.d. gives a nice looking curve. We will see why in today's class meeting.
Another Email Date: Sun, 28 Jan 2007 21:32:36 -0500 (EST) > Hey Professor Marron, I'm having some problems with C5. Ok, so I opened > excel spreadsheet, I typed in the mean and median, I did mean+ 3sd mean-3sd > For the X-value under NOrdmdist I put in the two x-values and then put in the > sd and mean, and put in FALSE for the cumulative since we want a height > distribution, but it gave me back a number. Am I doing the wrong function? Hmm, sounds like you may not be computing enough points to generate the plot. Basically you should generate a whole column of X-values, and then plug all of those into NORMDIST. An example of this available in Class Eg 8, which is linked to page 19 of the notes for 1/23/07. In that spreadsheet, this grid is in cells E78-E178. The corresponding calls of NORMDIST appear inthe range: J78 - J178. Then you plot those against each other.
Decision Problem: When should I do additional things in class? When should I send people to Open Tutorial Sessions? Depends on # benefitted • 1 or 2: send to Open Tutorials • Majority: should do in Class
Your Opinion? • Raise hand if you think this is worth class time right now. • Raise hand if you find this prospect boring, and want to move on instead. If we do this, go to Class Example 8: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg8Done.xls
Variable Relationships Chapter 2 in Text Idea: Look beyond single quantities, to how quantities relate to each other. E.g. How do HW scores “relate” to Exam scores? Section 2.1: Useful graphical device: Scatterplot
Plotting Bivariate Data Toy Example: (1,2) (3,1) (-1,0) (2,-1)
Plotting Bivariate Data Common Name: “Scatterplot” A look under the hood: EXCEL: Chart Wizard (colored bar icon) • Chart Type: XY (scatter) • Subtype controls points only, or lines • Later steps similar to above (can massage the pic!)
Important Aspects of Relations • Form of Relationship • Direction of Relationship • Strength of Relationship
I. Form of Relationship • Linear: Data approximately follow a line Previous Class Scores Example http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg10.xls Final vs. High values of HW is “best” • Nonlinear: Data follows different pattern Nice Example: Bralower’s Fossil Data http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg11.xls
Bralower’s Fossil Data http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg11.xls From T. Bralower, formerly of Geological Sci. Studies Global Climate, millions of years ago: • Ratios of Isotopes of Strontium • Reflects Ice Ages, via Sea Level (50 meter difference!) • As function of time • Clearly nonlinear relationship
II. Direction of Relationship • Positive Association (slopes upwards) X bigger Y bigger • Negative Association (slopes down) X bigger Y smaller E.g. X = alcohol consumption, Y = Driving Ability Clear negative association
III. Strength of Relationship Idea: How close are points to lying on a line? Revisit Class Scores Example: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg10.xls • Final Exam is “closely related to HW” • Midterm 1 less closely related to HW • Midterm 2 even related to Midterm 1
Linear Relationship HW 2.3, 2.5, 2.7, 2.11
Comparing Scatterplots Additional Useful Visual Tool: • Overlaying multiple data sets • Allows comparison • Use different colors or symbols • Easy in EXCEL (colors are automatic) Already done in HW scores example: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg12.xls
And Now for Something Completely Different Remember it takes a college degree to fly a plane, but only a high school diploma to fix one. After every flight, Qantas pilots fill out a form, called a gripe sheet which tells mechanics about problems with the aircraft. The mechanics correct the problems, document their repairs on the form, and then pilots review the gripe sheets before the next flight.
And Now for Something Completely Different Never let it be said that ground crews lack a sense of humor. Here are some actual maintenance complaints submitted by Qantas' pilots (marked with a P) and the solutions recorded (marked with an S) by maintenance engineers.
And Now for Something Completely Different Never let it be said that ground crews lack a sense of humor. Here are some actual maintenance complaints submitted by Qantas' pilots (marked with a P) and the solutions recorded (marked with an S) by maintenance engineers. By the way, Qantas is the only major airline that has never, ever, had an accident.
And Now for Something Completely Different P: Left inside main tire almost needs replacement. S: Almost replaced left inside main tire.
And Now for Something Completely Different P: Test flight OK, except auto-land very rough. S: Auto-land not installed on this aircraft.
And Now for Something Completely Different P: Dead bugs on windshield. S: Live bugs on back-order.
And Now for Something Completely Different P: Evidence of leak on right main landing gear. S: Evidence removed.
And Now for Something Completely Different P: IFF inoperative in OFF mode. S: IFF always inoperative in OFF mode.
And Now for Something Completely Different P: Number 3 engine missing. S: Engine found on right wing after brief search.
And Now for Something Completely Different P: Noise coming from under instrument panel. Sounds like a midget pounding on something with a hammer. S: Took hammer away from midget.
Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Context: • A numerical summary • In spirit of mean and standard deviation • But now applies to pairs of variables
Section 2.2: Correlation Main Idea: Quantify Strength of Relationship Specific Goals: • Near 1: for positive relat’ship & nearly linear • > 0: for positive relationship (slopes up) • = 0: for no relationship • < 0: for negative relationship (slopes down) • Near -1: for negative relat’ship & nearly linear
Correlation - Approach Numerical Approach: for symmetric around has similar properties Worked out Example : http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg13.xls
Correlation – Graphical View Plots (a) & (b), illustrating : • > 0 for positive relationship • < 0 for negative relationship • Bigger for data closer to line Problem 1: Not between -1 & 1 Problem 2: Feels “Scale”, see plot (c) Problem 3: Feels “Shift” even more, see (d) (even gets sign wrong!)
Correlation - Approach Solution to above problems: Standardize! Define Correlation
Correlation - Example Revisit above example http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg13.xls • r is always same, and ~1, for (a), (c), (d) • r < 0, and not so close to -1, for (b)
Correlation - Example A look under the hood http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg13.xls • Cols A&B: generated random numbers (will study later) • Product versions used SUMPRODUCT • r computed with CORREL (important) • r’s same for (a) & (c), since Y’s are “just shifted” • r’s also same for (d), since x’s and Y’s shifted (standardization cancels shifts & scales)
Correlation - Example Revisit Class Scores Example: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg10.xls • r is always > 0 • r is biggest for Final vs. HW • r is smallest for MT2 vs. MT1
Correlation - Example Fun Example from Publisher’s Website: http://bcs.whfreeman.com/ips5e/ Choose • Statistical Applets • Correlation and Regression Gives feeling for how correlation is affected by changing data.
Correlation - Example Fun Example from Publisher’s Website: http://bcs.whfreeman.com/ips5e/ Interesting Exercise: • Choose points to give correlation r = 0.95 (within 0.01) • Destroy with a few outliers
Correlation - HW HW: 2.23 2.25 2.27a
Correlation - Outliers Caution: Outliers can strongly affect correlation, r HW: 2.27b 2.30 (big outlier reduces correlation) Also: recompute correlation with outlier removed
And now for something completely different Recall Distribution of majors of students in this course:
And Now for Something Completely Different Tried to Google “Public Policy Jokes” But couldn’t find anything decent. Next tried “Public Health Jokes” And came up with…
And Now for Something Completely Different Regular Consumption of Guinness Well now, you see it's like this....
And Now for Something Completely Different A herd of buffalo can only move as fast as the slowest buffalo. And when the herd is hunted, it is the slowest and weakest ones at the rear that are killed. This natural selection is good for the herd as a whole because only the fittest survive thus improving the general health and speed of the entire herd.
And Now for Something Completely Different In much the same way the human brain only operates as quickly as the slowest of it's brain cells. Excessive intake of alcohol kills brain cells, as we all know, and naturally the alcohol attacks the slowest/weakest cells first....
And Now for Something Completely Different So it is as plain as the nose on your face that regular consumption of Guinness will eliminate the weaker, slower brain cells thus leaving the remaining cells the best in the brain.
And Now for Something Completely Different The end result, of course, is a faster more efficient brain. If you doubt this at all, tell me, isn't it true that we always feel a bit smarter after a few pints?
Section 2.3: Linear Regression Idea: Fit a line to data in a scatterplot • To learn about “basic structure” • To “model data” • To provide “prediction of new values”
Linear Regression Recall some basic geometry: A line is described by an equation: y = mx + b m = slope m b = y intercept b Varying m & b gives a “family of lines”, Indexed by “parameters” m & b
Basics of Lines Textbook’s notation: Y = bx + a b = m (above) = slope a = b (above) = y-intercept