6.6: Footnotes
- Page ID
- 374966
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Don’t freak out about the word ‘calculus’. We’re not doing derivatives and integrals here; we’re using that word in a generic sense, as in ‘a system for performing calculations’, or something like that. Also, don’t get freaked out about ‘mathematics’. This is really simple, fifth-grade stuff: adding and multiplying fractions and decimals. ↵
- Twain attributes the remark to British Prime Minister Benjamin Disraeli, though it’s not really clear who said it first. ↵
- If you think otherwise, you’re committing what’s known as the Gambler’s Fallacy. It’s surprisingly common. Go to a casino and you’ll see people committing it. Head to the roulette wheel, for example, where people can bet on whether the ball lands in a red or a black space. After a run of say, five reds in a row, somebody will commit the fallacy: “Red is hot! I’m betting on it again.” This person believes that the results of the previous spins somehow affect the probability of the outcome of the next one. But they don’t. Notice that an equally compelling (and fallacious) case can be made for black: “Five reds in a row? Black is due. I’m betting on black.” ↵
- A standard deck has 52 playing cards, equally divided among four suits (hearts, diamonds, clubs, and spades) with 13 different cards in each suit: Ace (A), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), and King (K). ↵
- This makes good sense. If you throw a coin twice, there are four distinct ways things could go: (1) you throw heads twice; (2) you throw heads the first time, tails the second; (3) you throw tails the first time, heads the second; (4) you throw tails twice. In three out of those four scenarios (all but the last), you’ve thrown at least one head. ↵
- Really: http://mentalfloss.com/article/62593...-weird-flavors ↵
- Inspired by an exercise from Copi and Cohen, pp. 596 - 597 ↵
- In this and what follows, I am indebted to Copi and Cohen’s presentation for inspiration. ↵
- It’s more complicated than this, but we’re simplifying to make things easier. ↵
- They’re insured through the FDIC—Federal Deposit Insurance Corporation—created during the Great Depression to prevent bank runs. Before this government insurance on deposits, if people thought a bank was in trouble, everybody tried to withdraw their money at the same time; that’s a “bank run”. Think about the scene in It’s a Wonderful Life when George is about to leave on his honeymoon, but he has to go back to the Bailey Building and Loan to prevent such a catastrophe. Anyway, if everybody knows they’ll get their money back even if the bank goes under, such things won’t happen; that’s what the FDIC is for. ↵
- Unless, of course, the federal government goes out of business. But in that case, ,000 is useful maybe as emergency toilet paper; I need canned goods and ammo at that point. ↵
- Again, there are all sorts of complications we’re glossing over to keep things simple. ↵
- Probably. There are different kinds of bankruptcies and lots of laws governing them; it’s possible for investors to get some money back in probate court. But it’s complicated. One thing’s for sure: our measly ,000 imaginary investment makes us too small-time to have much of a chance of getting paid during bankruptcy proceedings. ↵
- Historical data on the probability of default for companies at different ratings by agency are available. ↵
- Considerations like these are apparently the spark that lit the fuse on the financial crisis of late 2008. On September 15th of that year, the financial services firm Lehman Brothers filed for bankruptcy—the largest bankruptcy filing in history. The stock market went into a free-fall, and the economy ground to a halt. The problem was borrowing: companies couldn’t raise money in the usual way with corporate bonds. Such borrowing is the grease that keeps the engine of the economy running; without it, firms can’t fund their day-to-day operations. The reason companies couldn’t borrow was that investors were demanding too high a rate of interest. They were doing this because their personal estimations of P(paid) were all revised downward in the wake of Lehman’s bankruptcy: that was considered a reliable company to lend to; if they could go under, anybody could. ↵
- This function maps 1 unit of wealth to 10 units of utility (never mind what those units are). 2 units of wealth produces 30 units of utility, and so on: 3 – 48; 4 – 60; 5 – 70; 6 – 78; 7 – 84; 8 – 90; 9 – 96; 10 – 100. This mapping comes from Daniel Kahneman, 2011, Thinking, Fast and Slow, New York: Farrar, Strauss, and Giroux, p. 273. ↵
- For this and many other examples, see Kahneman 2011. ↵
- Again, see Kahneman 2011 for details. ↵
- There’s a whole literature on this. See this article for an overview: Hájek, Alan, "Interpretations of Probability", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/...ity-interpret/>. ↵
- It’s easy to derive this theorem, starting with the general product rule. We know P(E · H) = P(E) x P(H | E), no matter what ‘E’ and ‘H’ stand for. A little algebraic manipulation gives us P(H | E) = P(E · H) / P(E). It’s a truth of logic that the expression ‘E · H’ is equivalent to ‘H · E’, so we can replace ‘P(E · H)’ with ‘P(H · E)’ in the numerator. And again, by the general product rule, P(H · E) = P(H) x P(E | H)—our final numerator. ↵
- Provided you set things up carefully. Check out this video: https://www.youtube.com/watch?v=E43-CfukEgs. ↵
- Note: this is separate from the highly plausible claim that the Russians hacked e-mails from the Democratic National Committee and released them to the media before the election. ↵
- Here’s a representative rundown: http://www.dailykos.com/story/2016/1...on-NC-PA-WI-FL ↵
- Or, it is travelling a straight line, just through a space that is curved. Same thing. ↵
- I know. In the example, maybe it’s the cold weather and the new detergent causing my rash. Let’s set that possibility aside. ↵
- We add the subscript ‘k’ to the hypothesis we’re entertaining, and stipulate the k is between 1 and n simply to ensure that the hypothesis in question is among the set of exhaustive, mutually exclusive possibilities H1, H2, …, Hn. ↵
- Inspiration for this example, as with much that follows, comes from Darrell Huff, 1954, How to Lie with Statistics, New York: Norton. ↵
- In 2014, the richest fifth of American households accounted for over 51% of income; the poorest fifth, 3%. ↵
- “Gaussian” because the great German mathematician Carl Friedrich Gauss made a study of such distributions in the early 19th century (in connection with their relationship to errors in measurement). ↵
- This is a consequence of a mathematical result, the Central Limit Theorem, the basic upshot of which is that if some random variable (a trait like IQ, for example, to be concrete) is the sum of many independent random variables (causes of IQ differences: lots of different genetic factors, lots of different environmental factors), then the variable (IQ) will be normally distributed. The mathematical theorem deals with abstract numbers, and the distribution is only perfectly “normal” when the number of independent variables approaches infinity. That’s why real-life distributions are only approximately normal. ↵
- This is an exaggeration, of course, but not much of one. The average high in San Diego in January is 65°; in July, it’s 75°. Meanwhile, in Milwaukee, the average high in January is 29°, while in July it’s 80°. ↵
- Pick a person at random. How confident are you that they have an IQ between 70 and 130? 95.4%, that’s how confident. ↵
- As a matter of fact, in current practice, other confidence intervals are more often used: 90%, (exactly) 95%, 99%, etc. These ranges lie on either side of the mean within non-whole-number multiples of the standard deviation. For example, the exactly-95% interval is 1.96 SDs to either side of the mean. The convenience of calculators and spreadsheets to do our math for us makes these confidence intervals more practical. But we’ll stick with the 68.3/95.4/99.7 intervals for simplicity’s sake. ↵
- Again, the actual distribution may not be normal, but we will assume that it is in our examples. The basic patterns of reasoning are similar when dealing with different kinds of distributions. ↵
- Actually, the typical level is now exactly 95%, or 1.96 standard deviations from the mean. From now on, we’re just going to pretend that the 95.4% and 95% levels are the same thing. ↵
- Picture from a post at www.pregnancylab.net by David Grenache, PhD: http://www.pregnancylab.net/2012/11/...e-defects.html ↵
- False positive: the baby was perfectly healthy. ↵
- Except for those known to be at risk, who should start earlier. ↵
- Especially perverse are the cases in which the radiation treatment itself causes cancer in a patient who didn’t have to be treated to begin with. ↵
- PC Gøtzsche and KJ Jørgensen, 2013, Cochrane Database of Systematic Reviews (6), CD001877.pub5 ↵
- I am indebted for this example in particular (and for much background on the presentation of statistical reasoning in general) to John Norton, 1998, How Science Works, New York: McGraw-Hill, pp. 12.14 – 12.15. ↵
- And the mean (our result of .55). The mathematical details of the calculation needn’t detain us. ↵
- Here’s an actual survey with that result:http://angusreidglobal.com/wp-conten...3.04_Myths.pdf ↵
- Actually, it’s 3.1%, but never mind. ↵
- Interesting mathematical fact: these relationships hold no matter how big the general population from which you’re sampling (as long as it’s above a certain threshold). It could be the size of the population of Wisconsin or the population of China: if your sample is 600 Wisconsinites, your margin of error is 4%; if it’s 600 Chinese people, it’s still 4%. This is counterintuitive, but true—at least, in the abstract. We’re omitting the very serious difficulty that arises in actual polling (which we will discuss anon): finding the right 600 Wisconsinites or Chinese people to make your survey reliable; China will present more difficulty than Wisconsin. ↵
- It’s even harder than this paragraph makes it out to be. It’s usually impossible for a sample—the people you’ve talked to on the phone about the president or whatever—to mirror the demographics of the population exactly. So pollsters have to weight the responses of certain members of their sample more than others to make up for these discrepancies. This is more art than science. Different pollsters, presented with the exact same data, will make different choices about how to weight things, and will end up reporting different results. See this fascinating piece for an example: http://www.nytimes.com/interactive/2...bout.html?_r=0 ↵
- Source: http://www.nbcnews.com/politics/elec...-power-n301031 ↵
- http://spotlight.ipsos-na.com/index....ntial-support/ ↵
- See here, for example: https://www.washingtonpost.com/news/...=.f20212063a9c ↵
- See this study: https://www.ncbi.nlm.nih.gov/pubmed/21180247 ↵
- See this survey: http://www.gallup.com/poll/27847/Maj...Evolution.aspx ↵
- The title of this section, a lot of the topics it discusses, and even some of the examples it uses, are taken from Huff 1954. ↵
- This sounds good, but it’s bad macroeconomics. Most economists agree that during a downturn like that one, the government should borrow and spend more, not less, in order to stimulate the economy. The president knew this; he ushered a huge government spending bill through Congress (The American Reinvestment and Recovery Act) later that year. ↵
- This is a useful resource: http://www.slate.com/articles/news_a...er_weight.html ↵
- This example inspired by Huff 1954, pp. 106 - 107. ↵
- It may be as high as 2% for hand-counting! See here:https://www.sciencedaily.com/release...0202151713.htm ↵
- John Paulos, “We’re Measuring Bacteria with a Yardstick,” November 22, 2000, The New York Times. ↵
- Steven Verburg, “Study: Budget Could Hurt State’s Economy,” March 20, 2011, Wisconsin State Journal. ↵
- Not because the school’s administration was particularly enlightened. They could only open with the financial support of four wealthy women who made this a condition for their donations. ↵
- This example inspired by Huff 1954, pp. 110 - 111. ↵
- This example inspired by Huff 1954, pp. 77 - 79. ↵
- Liz Szabo, “Marijuana poses more risks than many realize,” July 27, 2014, USA Today. http://www.usatoday.com/story/news/n.../?sf29269095=1 ↵
- From German Lopez, “Marijuana sends more people to the ER than heroin. But that's not the whole story.” August 2, 2014, Vox.com. http://www.vox.com/2014/8/2/5960307/...roin-USA-Today ↵
- Alexander Hart, “Lying With Graphs, Republican Style (Now Featuring 50% More Graphs),” December 22, 2010, New Republic. https://newrepublic.com/article/7789...publican-style ↵
- Ezra Klein, “Lies, damn lies, and the 'Y' axis,” September 23, 2010, The Washington Post. http://voices.washingtonpost.com/ezr...he_y_axis.html ↵
- See here: http://www.wsj.com/articles/SB100014...67113524583554 ↵
- See here: https://en.wikipedia.org/wiki/Isotyp...ture_language) ↵
- I’ve been using this example in class for years, and something tells me I got it from somebody else’s book, but I’ve looked through all the books on my shelves and can’t find it. So maybe I made it up myself. But if I didn’t, this footnote acknowledges whoever did. (If you’re that person, let me know!) ↵
- See here: http://www.slate.com/blogs/moneybox/..._shortage.html ↵
- Source of image: https://en.wikipedia.org/wiki/Electo...United_States) ↵
- Ibid. ↵
- p. 103. ↵