Stat 31, Section 1, Last Time

Stat 31, Section 1, Last Time • T Distribution (handles unknown σ) • Computation with TDIST & TINV • Confidence Intervals • Hypothesis Tests

Reading In Textbook Approximate Reading for Today’s Material: Pages 485-504, 536-549 Approximate Reading for Next Class: Pages 555-566

Midterm II Coming on Tuesday, April 10 Think about: • Sheet of Formulas • Again single 8 ½ x 11 sheet • New, since now more formulas • Redoing HW… • Asking about those not understood • Will schedule Extra Office Hours

And now for somethingcompletely different… Professional Statisticians Dislike Excel: Very poor handling of numerics Unacceptable?!? Jeff Simonoff Example: http://www.stern.nyu.edu/~jsimonof/classes/1305/pdf/excelreg.pdf

And now for somethingcompletely different… A similar example: Class Example 28: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg28.xls Problem 1: Excel doesn’t keep enough significant digits (relative to other software) [single precision vs. double precision]

And now for somethingcompletely different… Surprise for me: Problem didn’t turn up… Potential reason: • Different Excel versions • I.e. Problem fixed in newest versions… • Check: Looked at my machine

And now for somethingcompletely different…

And now for somethingcompletely different… Quite different answers. Check Versions:

And now for somethingcompletely different… But organizational problems still exist… For me those were the major problems…

Comments on Email • Seems to be generally working well • People seem to be learning from it • But I got overwhelmed last night & this morning • Please try to ask questions earlier… Bad response from me: Problem 6.1

Bad Response: 6.61

Bad Response: 6.61 • My apologies • I tried to cut a corner • Didn’t follow personal rule of writing it out • Downside of mass communication… • Let’s fix this now • I will ask grader to accept everybody’s answer

Bad Response: 6.61 Problem: testing H0: mu = 1.4 Ha: mu > 1.4 Based on Z-score of 1.75

Bad Response: 6.61 Based on Z-score of 1.75 Good news: • ½ of work is already done • Don’t need to mess with Xbar, n, s.d…. • Just need P-value

Bad Response: 6.61 Based on Z-score of 1.75 P-value = P{what saw or m.c. | Bdry} = P{Z >= 1.75} = 1 – P{Z <= 1.75} = 1 - NORMDIST(1.75,0,1,true) = 0.0401 (Excel)

Bad Response: 6.61 P-value = 0.0401 Interpretation via “yes – no”: Compare 0.0401 to 0.05: Since 0.0401 < 0.05, say “have statistical significance” (this is answer to part (b))

Bad Response: 6.61 P-value = 0.0401 Part (c): “test at level 0.01” • I.e. compare P-value to 0.01 • Note 0.0401 > 0.01 • So do not have strong evidence • I.e. “Accept H0”

2 Sample Inference Main Idea: • Previously studied single populations • Modeled as: • Measurement Error • Counts • Did Inference: • Confidence Intervals • Hypothesis Tests

2 Sample Inference Main Idea: • Extend to two populations • Modeled as: • Measurement Error • Counts • Will Develop Inference: • Confidence Intervals • Hypothesis Tests

2 Sample Inference Location in Text: • Measurement Error • Sec. 7.1 • Sec. 7.2 • Counts • Sec. 8.2

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Important: Measurements Connected, e.g. made on same objects

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Approach: Apply 1 sample methods to:

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): Researchers studying Vitamin C in a product were concerned about loss of Vitamin C during shipment and storage. They marked a collection of bags at the factory, and measured the vitamin C. 5 months later, in Haiti, they found the same bags, and again measured the Vitamin C.

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): The data are the two Vitamin C measurements, made for each bag. • Set up hypotheses to examine the question of interest. • Perform the significance test, and summarize the result. • Find 95% CIs for the factory mean, and the Haiti mean, and the mean change.

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): a. Sample average difference = Some evidence factory is bigger, is it strong evidence??? Let = Difference: Haiti – Factory 1-sided, from “idea of loss”

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): b.

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): b. But recall how TDIST works: = So compute with:

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): • Excel Computation: Class Example 27, Part 3 http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg27.xls P-value = 1.87 x 10-5 Interpretation: very strong evidence either yes-no or gray level

2 Paired Samples Variations: • EXCEL function TTEST is useful here Notes: • Type is paired (discuss others later) • Get same answer from swapping Array 1 and Array 2 (check these in class example) • This is something Excel does well

2 Paired Samples Variations: • Can also use: Tools  Data Analysis  T-test Paired to give detailed results e.g. d.f. = 26 (others we haven’t learned yet)

2 Paired Samples E.g. Old Textbook 7.32 (now 7.39): • Confidence Intervals See Class Example 27, Part 3c http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg27.xls Margin of error = (same as above, but NORMINV  TINV) So CI has endpoints:

Paired Sampling CIs & Tests HW: 7.27, 7.31, 7.41 Interpret P-values: (i) yes-no (ii) gray-level (suggestion: use TTEST, for hypo tests)

And now for somethingcompletely different… Does the statement, “We've always done it like that” ring any bells? The US standard railroad gauge (distance between the rails) is 4 feet, 8.5 inches. That's an exceedingly odd number. Why was that gauge used?

And now for somethingcompletely different… Because that's the way they built them in England, and English expatriates built the US Railroads. Why did the English build them like that? Because the first rail lines were built by the same people who built the pre-railroad tramways, and that's the gauge they used.

And now for somethingcompletely different… Why did "they" use that gauge then? Because the people who built the tramways used the same jigs and tools that they used for building wagons, which used that wheel spacing.

And now for somethingcompletely different… Okay! Why did the wagons have that particular odd wheel spacing? Well, if they tried to use any other spacing, the wagon wheels would break on some of the old, long distance roads in England , because that's the spacing of the wheel ruts.

And now for somethingcompletely different… So who built those old rutted roads? Imperial Rome built the first long distance roads in Europe (and England ) for their legions. The roads have been used ever since. And the ruts in the roads? Roman war chariots formed the initial ruts, which everyone else had to match for fear of destroying their wagon wheels. Since the chariots were made for Imperial Rome , they were all alike in the matter of wheel spacing.

And now for somethingcompletely different… The United States standard railroad gauge of 4 feet, 8.5 inches is derived from the original specifications for an Imperial Roman war chariot. And bureaucracies live forever. So the next time you are handed a specification and wonder what horse's ass came up with it, you may be exactly right, because the Imperial Roman army chariots were made just wide enough to accommodate the back ends of two war horses!

And now for somethingcompletely different… When you see a Space Shuttle sitting on its launch pad, there are two big booster rockets attached to the sides of the main fuel tank. These are solid rocket boosters, or SRBs. The SRBs are made by Thiokol at their factory at Utah. The engineers who designed the SRBs would have preferred to make them a bit fatter, but the SRBs had to be shipped by train from the factory to the launch site.

And now for somethingcompletely different… The railroad line from the factory happens to run through a tunnel in the mountains. The SRBs had to fit through that tunnel. The tunnel is slightly wider than the railroad track, and the railroad track, as you now know, is about as wide as two horses' behinds. So, a major Space Shuttle design feature of what is arguably the world's most advanced transportation system was determined over two thousand years ago by the width of a horse's ass.

And now for somethingcompletely different… - And – you thought being a HORSE'S ASS wasn't important!

2 Sample Measurement Error Easy Case: Paired Differences Have Treatment 1: Treatment 2: Hard case: 2 different (unmatched) samples different!

2 Sample Measurement Error Hard case: 2 different (unmatched) samples Notes: • There are several variations • For Hypo. Testing, EXCEL works well • Variations well labelled in TTEST

2 Sample Measurement Error Hard case: 2 different (unmatched) samples Main Ideas: Data: Sample Averages:

2 Sample Measurement Error Hard case: 2 different (unmatched) samples Base inference on: Probability Theory (can show):

2 Sample Measurement Error Hard case: 2 different (unmatched) samples Probability Theory (can show): Assumptions: • Xs & Ys Independent • Otherwise based on Law of Averages

2 Sample Measurement Error Step towards statistical inference: 2 sample Z statistic • Just do standardization (usual idea) • Handle unknown s.d.s???

2 Sample Measurement Error For unknown s.d.s, use usual approx: For 2 sample t statistic

2 Sample Measurement Error 2 sample t statistic: Probability Distribution: • 2 sample version of t distribution • Well modelled by EXCEL using TTEST • Use this for Hypothesis Testing

Stat 31, Section 1, Last Time