MIT OpenCourseWare http://ocw.mit.edu 18.443 Statistics for Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Table from "Engineering toolbox" of air pressures (in five different units, last 5columns)at different elevations above (or below, if negative)sea level, measured in feet ormeters. feet meters inHg mmHg psia kgpcmsq kPa [1,] -5000 -1526 35.5800 903.70 17.480 1.2290 120.50 [2,] -4500 -1373 35.0000 889.00 17.190 1.2090 118.50 [3,] -4000 -1220 34.4200 874.30 16.900 1.1880 116.50 [4,] -3500 -1068 33.8400 859.50 16.620 1.1690 114.60 [5,] -3000 -915 33.2700 845.10 16.340 1.1490 112.70 [6,] -2500 -763 32.7000 830.60 16.060 1.1290 110.70 [7,] -2000 -610 32.1400 816.40 15.780 1.1090 108.80 [8,] -1500 -458 31.5800 802.10 15.510 1.0910 106.90 [9,] -1000 -305 31.0200 787.90 15.230 1.0710 105.00 [10,] -500 -153 30.4700 773.90 14.960 1.0520 103.10 [11,] 0 0 29.9200 760.00 14.696 1.0333 101.33 [12,] 500 153 29.3800 746.30 14.430 1.0150 [13,] 1000 305 28.8600 733.00 14.160 0.9560 [14,] 1500 458 28.3300 719.60 13.910 0.9780 [15,] 2000 610 27.8200 706.60 13.660 0.9600 [16,] 2500 763 27.3200 693.90 13.410 0.9430 [17,] 3000 915 26.8200 681.20 13.170 0.9260 [18,] 3500 1068 26.3300 668.80 12.930 0.9090 [19,] 4000 1220 25.8400 656.30 12.690 0.8920 [20,] 4500 1373 25.3700 644.40 12.460 0.8760 [21,] 5000 1526 24.9000 632.50 12.230 0.8600 [22,] 6000 1831 23.9900 609.30 11.780 0.8280 [23,] 7000 2136 23.1000 586.70 11.340 0.7970 [24,] 8000 2441 22.2300 564.60 10.910 0.7670 [25,] 9000 2746 21.3900 543.30 10.500 0.7380 [26,] 10000 3050 20.5800 522.70 10.100 0.7100 [27,] 15000 4577 16.8900 429.00 8.290 0.5830 [28,] 20000 6102 13.7600 349.50 6.760 0.4750 [29,] 25000 7628 11.1200 282.40 5.460 0.3840 99.49 97.63 95.91 94.19 92.46 90.81 89.15 87.49 85.91 84.33 81.22 78.19 75.22 72.40 69.64 57.16 46.61 37.65 30.13 23.93 18.82 14.82 11.65 9.17 7.24 4.49 2.80 1.76 1.12 [30,] 30000 9153 [31,] 35000 10679 [32,] 40000 12204 [33,] 45000 13730 [34,] 50000 15255 [35,] 55000 16781 [36,] 60000 18306 [37,] 70000 21357 [38,] 80000 24408 [39,] 90000 27459 [40,] 100000 30510 8.9030 226.10 4.370 0.3070 7.0600 179.30 3.470 0.2440 5.5580 141.20 2.730 0.1920 4.3750 111.10 2.150 0.1510 3.4440 87.50 1.690 0.1190 2.7120 68.90 1.330 0.0940 2.1350 54.20 1.050 0.0740 1.3250 33.70 0.650 0.0460 0.8273 21.00 0.410 0.0290 0.5200 13.20 0.260 0.0180 0.3290 8.36 0.160 0.0110p. 2. Air pressure as a function of elevation, from the "Engineering toolbox" data: The elevations in "feet" and "meter[s]" columns are the sameexcept for units. The "feet" are evidently design points, beinground numbers, not random variables. Actual measured air pressuresare random variables, depending on weather as well as elevation,but in the air pressures as given it seems that weather effectshave been averaged out after multiple measurements, and the pressurein mmHg (mm. mercury) is standardized as 760.00 at sea level.Let's see if we can find how air pressure depends on elevation viaregression. First, let's look at simple linear regression. Regressionscan be done by R (you aren't required to use R in PS3, however). Here's the output, where "lm" indicates we're doing a linearregression, of "mmHg" (column header for air pressure in mmHgunits) on "feet" of elevation, and "airtable" is the name I gavethe table, also in R. For simple linear regression, "Multiple R-squared"is just the square of the ordinary correlation coefficient, so r =-0.9284 (it's negative because air pressure decreases with elevation).The most negative residual is < -185 and the largest is > 290 --- toobig. In the "Coefficients" table, the last three columns are based onassuming the normal model. The large t and F statistics show thecoefficients are both very significantly different from 0, assumingthe normal model for errors. On the next page we'll look at theresiduals from the model for a pattern. Here's the R output:------Call: lm(formula = mmHg ~ feet, data = airtable) Residuals: Min 1Q Median 3Q Max -185.169 -78.586 6.451 81.254 290.234 Coefficients: Estimate Std. Error t value Pr(>|t|)(Intercept) 7.083e+02 2.182e+01 32.46 <2e-16 *** feet -9.902e-03 6.431e-04 -15.40 <2e-16 *** ---Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 114.7 on 38 degrees of freedomMultiple R-Squared: 0.8619, Adjusted R-squared: 0.8582F-statistic: 237.1 on 1 and 38 DF, p-value: < 2.2e-16Air pressure as a function of height, p. 3: residuals from the simple linear regression ofpressure (in mmHg) on height. -------[1,] [2,] [3,] [4,] [5,] [6,] 145.859857 136.110876 126.361895 116.512913 107.063932 97.514951 [7,] [8,] [9,] [10,] [11,] [12,] 88.265970 78.916988 69.668007 60.619026 51.670044 42.921063 [13,] [14,] [15,] [16,] [17,] [18,] 34.572082 26.123100 18.074119 10.325138 2.576157 -4.872825 [19,] [20,] [21,] [22,] [23,] [24,] -12.421806 -19.370787 -26.319769 -39.617731 -52.315694 -64.513657 [25,] [26,] [27,] [28,] [29,] [30,] -75.911619 -86.609582 -130.799395 -160.789208 -178.379021 -185.168834 [31,] [32,] [33,] [34,] [35,] [36,] -182.458647 -171.048460 -151.638273 -125.728086 -94.817899 -60.007713 [37,] [38,] [39,] [40,] 18.512661 104.833035 196.053409 290.233783 ----- These residuals show a very strong convex pattern, namely they start out positive and decreasing, become negative while still decreasing, reach a minimum at the 30th residual, then start to increase, eventually becoming positive and increasing. There is only one "turning point" where the residuals change between decreasing and increasing. This is very incompatible with i.i.d. residuals (normal or otherwise) so we'll consider other regressions.18.443 February 18, 2009, p. 4 I tried a quadratic regression, mmHg = beta_0 + beta_1 h + beta_2 h^2 where h is elevation in feet and got the following results from R. Here in R's somewhat cryptic notation "I" is just the identity function, but it's required by the system, and "+" just means both h (in feet) and h^2 are regressors, not that they're added. ---- Call: lm(formula = mmHg ~ feet + I(feet^2), data = airtable) Residuals: Min 1Q Median 3Q Max -73.629 -20.720 -1.802 23.558 52.166 Coefficients: Estimate Std. Error t value Pr(>|t|)(Intercept) 7.543e+02 5.986e+00 126.00 <2e-16 *** feet