Linear Algebra

Topic: Line of Best Fit 265We close by considering the times for the men’s mile race [Oakley & Baker].These are the world records that were in force on January first of the givenyears. We want to project when a 3:40 mile will be run.year 1870 1880 1890 1900 1910 1920 1930seconds 268.8 264.5 258.4 255.6 255.6 252.6 250.41940 1950 1960 1970 1980 1990 2000246.4 241.4 234.5 231.1 229.0 226.3 223.1We can see below that the data is surprisingly linear. With this input⎛ ⎞ ⎛ ⎞1 1860280.01 1870268.8A =⎜..⃗v =⎟ ⎜ .⎟⎝1 1990⎠⎝226.3⎠1 2000223.1the Python program at this Topic’s end gives slope = −0.35 and intercept =925.53 (rounded to two places; the original data is good to only about a quarterof a second since much of it was hand-timed).seconds2602402201870 1890 1910 1930 1950 1970 1990yearWhen will a 220 second mile be run? Solving the equation of the line of best fitgives an estimate of the year 2008.This example is amusing, but serves as a caution — obviously the linearityof the data will break down someday (as indeed it does prior to 1860).ExercisesThe calculations here are best done on a computer. In addition, some of the problemsrequire more data, available in your library, on the net, or in the Answers tothe Exercises.1 Use least-squares to judge if the coin in this experiment is fair.flips 8 16 24 32 40heads 4 9 13 17 202 For the men’s mile record, rather than give each of the many records and itsexact date, we’ve “smoothed” the data somewhat by taking a periodic sample. Dothe longer calculation and compare the conclusions.3 Find the line of best fit for the men’s 1500 meter run. How does the slopecompare with that for the men’s mile? (The distances are close; a mile is about1609 meters.)