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MIT 18 443 - HOMEWORK -18.443

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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.18.443 problems dalist jumped the regression seems (b) Do the regression and 1. In the Olympics in the given years, the men’s long jump gold mefollowing distances (in cm.): year distance (cm.) year distance (cm.) 1952 758.3 1972 825.7 1956 784.4 1976 835.9 1960 813.6 1980 855.6 1964 808.5 1984 855.6 1968 891.9 (a) Taking x as the year and y as the winning distance, what kind of suitable here: y-on-x, x-on-y, or bfsd? evaluate the residuals.(c) Is there anything remarkable about the residuals? If anything, what? 2. The following data were reported on windmill power output at various wind speeds. The units of the (DC) power output are not given by the source where the data were found. For this problem data are selected so that the speeds look more like design p oints than they would have in the full data. output speed (mph) 0.123 2.45 0.558 3.05 1.194 4.10 1.582 5.00 1.822 6.00 1.800 7.00 2.166 8.15 2.303 9.10 2.386 9.70 2.236 10.00 (a) Letting y be power output and x wind speed, what kind of linear regression, if any, y-on-x, x-on-y, bfsd, seems appropriate for these data? Hints: for y-on-x, and perhaps also for bfsd, what values of power output would linear regression predict for windspeeds below, or above, the range shown, and would t hese be realistic? Would anyone be interested in “predicting” wind speed from power output? (b) Consider the range of wind speeds from 3 to 9.1 mph, which contains 7 of the 10 given observations. Find the correlation of wind speed and power output for these 7, and al so do the y-on-x regression for them. Also find the correlation for all 10 observations and compare it. 3. Data about average air pressure at given elevations from “The Engineering Toolbox” are given in a handout, which on the course website, subdirectory Airpressure, is a sequence of seven 1-page text files “airps1,...,airps7.” 1Since the el evations in feet are round numbers it seems that they are design point s. The last five columns are pressures in various units, linearly related to each o ther except for rounding error. So we can concentrat e on the two columns “feet” of elevatio n and “mmHg” which is pressure, measured in millimeters of mercury, averaged over weather conditions, and set at 760.0 at sea level for standardization. Regressions for all row s are described in the handout. For t his problem set, consider only t hose rows where the elevati o n in feet is 5, 000k for k = 0, 1, . .., 8, i.e. from sea level up to 40,000 feet in intervals of 5,000 feet. Do the following carefully, as the results will be used in the next problem. (a) For the given nine elevations, do a simple l inear regression of pressure (in mmHg) on elevati on (in feet). Give the slope and intercept, and the correlation. (b) Find the residuals. Do they show a pattern indicating whether linear regression fits the data well? 4. If one were to do a simple linear regression of the residuals on elevation, both the intercept and slo pe should be 0 if the regression wa s done correctly. (a) Do a simple linear regression of the residuals o n the square of elevation. Give the intercept and the coefficient (of elevation squared, which is not a slope per se). Also give the correlation coefficient. (This together with Problem 3 i s not the same as doing a quadratic regression of pressure on height (feet) and its square.) (b) Find the residuals in this new regression. Now is there any pattern indicating whether a quadratic model fits the data? (c) If p is pressure and h is elevation, consider general polynomial models of the form pj = Q(hj ) + ǫj where Q i s a polynomial of gi ven degree (such as 1 in problem 3 and 2 in parts (a) and (b) of this problem), and ǫj are i.i.d. N(0, σ2). Can any such model work well for all positive values of h? Hint: consider very large values of h (say beyond the stratosphere). 5. For the same data again: (a) Do a simple linear regression of the (natural) logarithm of the pressure ln p on the elevati on h. Give the intercept ˆa and the slope ˆb. Also give the correlation. How does it compare to those in the previous two problems? (b) From (a), solving for p g ives pj = ea+bhj + ǫj . A s in problem 4(c), does such a model give absurd val ues of p for very large values of h, in terms of the expected value of p given h? (c) Suppose that the residuals i n part (a) show a pattern similar to that for the full data set, shown in the last page of the handout. Find the residuals for part (b). What pattern if any do they show? (d) Does the variance of the measurements of air pressure, in the range from 5,000 to 40,000 feet, seem to depend on the elevation? What about the variance in the log of p? Hint: do the residuals seem to be on a fairly smoot h curve as opposed to being random? If so, then the measurements t hemselves would be on a smooth curve. There is at least rounding error in them, so concentrate on that. Does that seem to change with elevation? What about the resulting errors in log p?


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