Linear Algebra/Topic: Line of Best Fit/Solutions

Solutions

Problem 1

Use least-squares to judge if the coin in this experiment is fair.

flips	8	16	24	32	40
heads	4	9	13	17	20

Answer

As with the first example discussed above, we are trying to find a best $m$ to "solve" this system.

${\begin{array}{*{5}{rc}r}8m&=&4\\16m&=&9\\24m&=&13\\32m&=&17\\40m&=&20\end{array}}$

Projecting into the linear subspace gives this

${\frac {{\begin{pmatrix}4\\9\\13\\17\\20\end{pmatrix}}\cdot {\begin{pmatrix}8\\16\\24\\32\\40\end{pmatrix}}}{{\begin{pmatrix}8\\16\\24\\32\\40\end{pmatrix}}\cdot {\begin{pmatrix}8\\16\\24\\32\\40\end{pmatrix}}}}\cdot {\begin{pmatrix}8\\16\\24\\32\\40\end{pmatrix}}={\frac {1832}{3520}}\cdot {\begin{pmatrix}8\\16\\24\\32\\40\end{pmatrix}}$

so the slope of the line of best fit is approximately $0.52$.

Problem 2

For the men's mile record, rather than give each of the many records and its exact date, we've "smoothed" the data somewhat by taking a periodic sample. Do the longer calculation and compare the conclusions.

Answer

With this input

$A={\begin{pmatrix}1&1852.71\\1&1858.88\\\vdots &\vdots \\1&1985.54\\1&1993.71\end{pmatrix}}\qquad b={\begin{pmatrix}292.0\\285.0\\\vdots \\226.32\\224.39\end{pmatrix}}$

(the dates have been rounded to months, e.g., for a September record, the decimal $.71\approx (8.5/12)$ was used), Maple responded with an intercept of $b=994.8276974$ and a slope of $m=-0.3871993827$.

Problem 3

Find the line of best fit for the men's $1500$ meter run. How does the slope compare with that for the men's mile? (The distances are close; a mile is about $1609$ meters.)

Answer

With this input (the years are zeroed at $1900$)

$A:={\begin{pmatrix}1&.38\\1&.54\\\vdots \vdots \\1&92.71\\1&95.54\end{pmatrix}}\qquad b={\begin{pmatrix}249.0\\246.2\\\vdots \\208.86\\207.37\end{pmatrix}}$

(the dates have been rounded to months, e.g., for a September record, the decimal $.71\approx (8.5/12)$ was used), Maple gives an intercept of $b=243.1590327$ and a slope of $m=-0.401647703$. The slope given in the body of this Topic for the men's mile is quite close to this.

Problem 4

Find the line of best fit for the records for women's mile.

Answer

With this input (the years are zeroed at $1900$)

$A={\begin{pmatrix}1&21.46\\1&32.63\\\vdots &\vdots \\1&89.54\\1&96.63\end{pmatrix}}\qquad b={\begin{pmatrix}373.2\\327.5\\\vdots \\255.61\\252.56\end{pmatrix}}$

(the dates have been rounded to months, e.g., for a September record, the decimal $.71\approx (8.5/12)$ was used), MAPLE gave an intercept of $b=378.7114894$ and a slope of $m=-1.445753225$.

Problem 5

Do the lines of best fit for the men's and women's miles cross?

Answer

These are the equations of the lines for men's and women's mile (the vertical intercept term of the equation for the women's mile has been adjusted from the answer above, to zero it at the year $0$, because that's how the men's mile equation was done).

${\begin{array}{rl}y&=994.8276974-0.3871993827x\\y&=3125.6426-1.445753225x\end{array}}$

Obviously the lines cross. A computer program is the easiest way to do the arithmetic: MuPAD gives $x=2012.949004$ and $y=215.4150856$ ($215$ seconds is $3$ minutes and $35$ seconds). Remark. Of course all of this projection is highly dubious — for one thing, the equation for the women is influenced by the quite slow early times — but it is nonetheless fun.

Problem 6

When the space shuttle Challenger exploded in 1986, one of the criticisms made of NASA's decision to launch was in the way the analysis of number of O-ring failures versus temperature was made (of course, O-ring failure caused the explosion). Four O-ring failures will cause the rocket to explode. NASA had data from 24 previous flights.

temp °F	53	75	57	58	63	70	70	66	67	67	67
failures	3	2	1	1	1	1	1	0	0	0	0
temp °F	68	69	70	70	72	73	75	76	76	78	79	80	81
failures	0	0	0	0	0	0	0	0	0	0	0	0	0

The temperature that day was forecast to be $31^{\circ }{\text{F}}$.

1. NASA based the decision to launch partially on a chart showing only the flights that had at least one O-ring failure. Find the line that best fits these seven flights. On the basis of this data, predict the number of O-ring failures when the temperature is $31$, and when the number of failures will exceed four.
2. Find the line that best fits all 24 flights. On the basis of this extra data, predict the number of O-ring failures when the temperature is $31$, and when the number of failures will exceed four.

Which do you think is the more accurate method of predicting? (An excellent discussion appears in (Dalal, Folkes & Hoadley 1989).)

Answer

1. A computer algebra system like MAPLE or MuPAD will give an intercept of $b=4259/1398\approx 3.239628$ and a slope of $m=-71/2796\approx -0.025393419$ Plugging $x=31$ into the equation yields a predicted number of O-ring failures of $y=2.45$ (rounded to two places). Plugging in $y=4$ and solving gives a temperature of $x=-29.94^{\circ }$F.
2. On the basis of this information

   $A={\begin{pmatrix}1&53\\1&75\\\vdots \\1&80\\1&81\end{pmatrix}}\qquad b={\begin{pmatrix}3\\2\\\vdots \\0\\0\end{pmatrix}}$

   MAPLE gives the intercept $b=187/40=4.675$ and the slope $m=-73/1200\approx -0.060833$. Here, plugging $x=31$ into the equation predicts $y=2.79$ O-ring failures (rounded to two places). Plugging in $y=4$ failures gives a temperature of $x=11^{\circ }$F.

   

Problem 7

This table lists the average distance from the sun to each of the first seven planets, using earth's average as a unit.

Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus
0.39	0.72	1.00	1.52	5.20	9.54	19.2

1. Plot the number of the planet (Mercury is $1$, etc.) versus the distance. Note that it does not look like a line, and so finding the line of best fit is not fruitful.
2. It does, however look like an exponential curve. Therefore, plot the number of the planet versus the logarithm of the distance. Does this look like a line?
3. The asteroid belt between Mars and Jupiter is thought to be what is left of a planet that broke apart. Renumber so that Jupiter is $6$, Saturn is $7$, and Uranus is $8$, and plot against the log again. Does this look better?
4. Use least squares on that data to predict the location of Neptune.
5. Repeat to predict where Pluto is.
6. Is the formula accurate for Neptune and Pluto?

This method was used to help discover Neptune (although the second item is misleading about the history; actually, the discovery of Neptune in position $9$ prompted people to look for the "missing planet" in position $5$). See (Gardner 1970)

Answer

1. The plot is nonlinear.

   

2. Here is the plot.

   

   There is perhaps a jog up between planet $4$ and planet $5$.

3. This plot seems even more linear.

   

4. With this input

   $A={\begin{pmatrix}1&1\\1&2\\1&3\\1&4\\1&6\\1&7\\1&8\end{pmatrix}}\qquad b={\begin{pmatrix}-0.40893539\\-0.1426675\\0\\0.18184359\\0.71600334\\0.97954837\\1.2833012\end{pmatrix}}$

   MuPAD gives that the intercept is $b=-0.6780677466$ and the slope is $m=0.2372763818$.
5. Plugging $x=9$ into the equation $y=-0.6780677466+0.2372763818x$ from the prior item gives that the log of the distance is $1.4574197$, so the expected distance is $28.669472$. The actual distance is about $30.003$.
6. Plugging $x=10$ into the same equation gives that the log of the distance is $1.6946961$, so the expected distance is $49.510362$. The actual distance is about $39.503$.

Problem 8

William Bennett has proposed an Index of Leading Cultural Indicators for the US (Bennett 1993). Among the statistics cited are the average daily hours spent watching TV, and the average combined SAT scores.

	1960	1965	1970	1975	1980	1985	1990	1992
TV	5:06	5:29	5:56	6:07	6:36	7:07	6:55	7:04
SAT	975	969	948	910	890	906	900	899

Suppose that a cause and effect relationship is proposed between the time spent watching TV and the decline in SAT scores (in this article, Mr. Bennett does not argue that there is a direct connection).

1. Find the line of best fit relating the independent variable of average daily TV hours to the dependent variable of SAT scores.
2. Find the most recent estimate of the average daily TV hours (Bennett's cites Neilsen Media Research as the source of these estimates). Estimate the associated SAT score. How close is your estimate to the actual average? (Warning: a change has been made recently in the SAT, so you should investigate whether some adjustment needs to be made to the reported average to make a valid comparison.)

Answer

1. With this input

   $A={\begin{pmatrix}1&306\\1&329\\1&356\\1&367\\1&396\\1&427\\1&415\\1&424\end{pmatrix}}\qquad b={\begin{pmatrix}975\\969\\948\\910\\890\\906\\900\\899\end{pmatrix}}$

   MAPLE gives the intercept $b=34009779/28796\approx 1181.0591$ and the slope $m=-19561/28796\approx -0.6793$.

   

Additional Data

Data on the progression of the world's records (taken from the Runner's World web site) is below.

Progression of Men's Mile Record
time	name	date
4:52.0	Cadet Marshall (GBR)	02Sep52
4:45.0	Thomas Finch (GBR)	03Nov58
4:40.0	Gerald Surman (GBR)	24Nov59
4:33.0	George Farran (IRL)	23May62
4:29 3/5	Walter Chinnery (GBR)	10Mar68
4:28 4/5	William Gibbs (GBR)	03Apr68
4:28 3/5	Charles Gunton (GBR)	31Mar73
4:26.0	Walter Slade (GBR)	30May74
4:24 1/2	Walter Slade (GBR)	19Jun75
4:23 1/5	Walter George (GBR)	16Aug80
4:19 2/5	Walter George (GBR)	03Jun82
4:18 2/5	Walter George (GBR)	21Jun84
4:17 4/5	Thomas Conneff (USA)	26Aug93
4:17.0	Fred Bacon (GBR)	06Jul95
4:15 3/5	Thomas Conneff (USA)	28Aug95
4:15 2/5	John Paul Jones (USA)	27May11
4:14.4	John Paul Jones (USA)	31May13
4:12.6	Norman Taber (USA)	16Jul15
4:10.4	Paavo Nurmi (FIN)	23Aug23
4:09 1/5	Jules Ladoumegue (FRA)	04Oct31
4:07.6	Jack Lovelock (NZL)	15Jul33
4:06.8	Glenn Cunningham (USA)	16Jun34
4:06.4	Sydney Wooderson (GBR)	28Aug37
4:06.2	Gunder Hagg (SWE)	01Jul42
4:04.6	Gunder Hagg (SWE)	04Sep42
4:02.6	Arne Andersson (SWE)	01Jul43
4:01.6	Arne Andersson (SWE)	18Jul44
4:01.4	Gunder Hagg (SWE)	17Jul45
3:59.4	Roger Bannister (GBR)	06May54
3:58.0	John Landy (AUS)	21Jun54
3:57.2	Derek Ibbotson (GBR)	19Jul57
3:54.5	Herb Elliott (AUS)	06Aug58
3:54.4	Peter Snell (NZL)	27Jan62
3:54.1	Peter Snell (NZL)	17Nov64
3:53.6	Michel Jazy (FRA)	09Jun65
3:51.3	Jim Ryun (USA)	17Jul66
3:51.1	Jim Ryun (USA)	23Jun67
3:51.0	Filbert Bayi (TAN)	17May75
3:49.4	John Walker (NZL)	12Aug75
3:49.0	Sebastian Coe (GBR)	17Jul79
3:48.8	Steve Ovett (GBR)	01Jul80
3:48.53	Sebastian Coe (GBR)	19Aug81
3:48.40	Steve Ovett (GBR)	26Aug81
3:47.33	Sebastian Coe (GBR)	28Aug81
3:46.32	Steve Cram (GBR)	27Jul85
3:44.39	Noureddine Morceli (ALG)	05Sep93
3:43.13	Hicham el Guerrouj (MOR)	07Jul99

Progression of Men's 1500 Meter Record
time	name	date
4:09.0	John Bray (USA)	30May00
4:06.2	Charles Bennett (GBR)	15Jul00
4:05.4	James Lightbody (USA)	03Sep04
3:59.8	Harold Wilson (GBR)	30May08
3:59.2	Abel Kiviat (USA)	26May12
3:56.8	Abel Kiviat (USA)	02Jun12
3:55.8	Abel Kiviat (USA)	08Jun12
3:55.0	Norman Taber (USA)	16Jul15
3:54.7	John Zander (SWE)	05Aug17
3:53.0	Paavo Nurmi (FIN)	23Aug23
3:52.6	Paavo Nurmi (FIN)	19Jun24
3:51.0	Otto Peltzer (GER)	11Sep26
3:49.2	Jules Ladoumegue (FRA)	05Oct30
3:49.0	Luigi Beccali (ITA)	17Sep33
3:48.8	William Bonthron (USA)	30Jun34
3:47.8	Jack Lovelock (NZL)	06Aug36
3:47.6	Gunder Hagg (SWE)	10Aug41
3:45.8	Gunder Hagg (SWE)	17Jul42
3:45.0	Arne Andersson (SWE)	17Aug43
3:43.0	Gunder Hagg (SWE)	07Jul44
3:42.8	Wes Santee (USA)	04Jun54
3:41.8	John Landy (AUS)	21Jun54
3:40.8	Sandor Iharos (HUN)	28Jul55
3:40.6	Istvan Rozsavolgyi (HUN)	03Aug56
3:40.2	Olavi Salsola (FIN)	11Jul57
3:38.1	Stanislav Jungwirth (CZE)	12Jul57
3:36.0	Herb Elliott (AUS)	28Aug58
3:35.6	Herb Elliott (AUS)	06Sep60
3:33.1	Jim Ryun (USA)	08Jul67
3:32.2	Filbert Bayi (TAN)	02Feb74
3:32.1	Sebastian Coe (GBR)	15Aug79
3:31.36	Steve Ovett (GBR)	27Aug80
3:31.24	Sydney Maree (usa)	28Aug83
3:30.77	Steve Ovett (GBR)	04Sep83
3:29.67	Steve Cram (GBR)	16Jul85
3:29.46	Said Aouita (MOR)	23Aug85
3:28.86	Noureddine Morceli (ALG)	06Sep92
3:27.37	Noureddine Morceli (ALG)	12Jul95
3:26.00	Hicham el Guerrouj (MOR)	14Jul98

Progression of Women's Mile Record
time	name	date
6:13.2	Elizabeth Atkinson (GBR)	24Jun21
5:27.5	Ruth Christmas (GBR)	20Aug32
5:24.0	Gladys Lunn (GBR)	01Jun36
5:23.0	Gladys Lunn (GBR)	18Jul36
5:20.8	Gladys Lunn (GBR)	08May37
5:17.0	Gladys Lunn (GBR)	07Aug37
5:15.3	Evelyne Forster (GBR)	22Jul39
5:11.0	Anne Oliver (GBR)	14Jun52
5:09.8	Enid Harding (GBR)	04Jul53
5:08.0	Anne Oliver (GBR)	12Sep53
5:02.6	Diane Leather (GBR)	30Sep53
5:00.3	Edith Treybal (ROM)	01Nov53
5:00.2	Diane Leather (GBR)	26May54
4:59.6	Diane Leather (GBR)	29May54
4:50.8	Diane Leather (GBR)	24May55
4:45.0	Diane Leather (GBR)	21Sep55
4:41.4	Marise Chamberlain (NZL)	08Dec62
4:39.2	Anne Smith (GBR)	13May67
4:37.0	Anne Smith (GBR)	03Jun67
4:36.8	Maria Gommers (HOL)	14Jun69
4:35.3	Ellen Tittel (FRG)	20Aug71
4:34.9	Glenda Reiser (CAN)	07Jul73
4:29.5	Paola Pigni-Cacchi (ITA)	08Aug73
4:23.8	Natalia Marasescu (ROM)	21May77
4:22.1	Natalia Marasescu (ROM)	27Jan79
4:21.7	Mary Decker (USA)	26Jan80
4:20.89	Lyudmila Veselkova (SOV)	12Sep81
4:18.08	Mary Decker-Tabb (USA)	09Jul82
4:17.44	Maricica Puica (ROM)	16Sep82
4:15.8	Natalya Artyomova (SOV)	05Aug84
4:16.71	Mary Decker-Slaney (USA)	21Aug85
4:15.61	Paula Ivan (ROM)	10Jul89
4:12.56	Svetlana Masterkova (RUS)	14Aug96

References

• Bennett, William (March 15, 1993), "Quantifying America's Decline", Wall Street Journal{{citation}}: CS1 maint: date and year (link)
• Dalal, Siddhartha; Folkes, Edward; Hoadley, Bruce (Fall 1989), "Lessons Learned from Challenger: A Statistical Perspective", Stats: the Magazine for Students of Statistics, pp. 14–18{{citation}}: CS1 maint: date and year (link)
• Gardner, Martin (April 1970), "Mathematical Games, Some mathematical curiosities embedded in the solar system", Scientific American, pp. 108–112{{citation}}: CS1 maint: date and year (link)