Description of 'forbes.dat' (free format) variable ----------------------------------------- Source 1=Forbes 2=Hooker T Temperature (degrees Farenheit) P Pressure(inches of Hg) Forbes' data on Boiling Point of Water and Barometric Pressure In the 1840's and 1850's a Scottish physicist, James T Forbes, wanted to be able to estimate altitude above sea level from measurements of the boiling point of water. He knew that altitude could be determined from barometric pressure, measured with a barometer, with lower pressures corresponding to higher altitudes. In the experiment discussed here, he studied the relationship between pressure and boiling point. His interest in this problem was motivated by the difficulty in transporting the fragile barometers of the 1940s. Measuring the boiling point would give travelers a quick way of estimating altitudes. Forbes collected data in the Alps and in Scotland. After choosing a location, he assembled his apparatus, and measured pressure and boiling point. Pressure measurements were recorded in inches of mercury, adjusted for the difference between the ambient air temperature when he took the measurements and a standard temperature. Boiling point was measured in degrees Fahrenheit. The data for n = 17 locales are reproduced from an 1857 paper (Forbes, 1857) T(¡F) P(in Hg) 194.5 20.79 194.3 20.79 197.9 22.40 198.4 22.67 199.4 23.15 199.9 23.35 200.9 23.89 201.1 23.99 201.4 24.02 201.3 24.01 203.6 25.14 204.6 26.57 209.5 28.49 208.6 27.76 210.7 29.04 211.9 29.88 212.2 30.06 On reviewing the data, there are several questions of potential interest. How are pressure and boiling point related? Is the relationship strong or weak? Can we predict pressure from temperature, and if so, how well? Forbes' theory suggested that over the range of observed values the graph of boiling point versus the logarithm of pressure yields a straight line. 1.2 Hooker's Data In his paper on boiling points and temperatures, Forbes also presented data on the same two quantities by Dr James Hooker. Unlike Forbes, however, Hooker took his measurements in the Himalaya Mountains, generally at higher altitudes. Hooker's n=31 pairs of measurements on T = the boiling point of water (degrees Fahrenheit) and P = corrected barometric pressure (inches of mercury) are given below. T(¡F) P(in Hg) T(¡F) P(in Hg) 210.8 29.211 210.2 28.559 208.4 27.972 202.5 24.697 200.6 23.726 200.1 23.369 199.5 23.030 197.0 21.892 196.4 21.928 196.3 21.654 195.6 21.605 193.4 20.480 193.6 20.212 191.4 19.758 191.1 19.490 190.6 19.386 189.5 18.869 188.8 18.356 188.5 18.507 185.7 17.267 186.0 17.221 185.6 17.062 184.1 16.959 184.6 16.881 184.1 16.817 183.2 16.385 182.4 16.235 181.9 16.106 181.9 15.928 181.0 15.919 180.6 15.376 EXERCISES: 1.2.1 Draw the scatter plot of P versus T. Would a straight line closely match the data? (A graph of the residuals from the regression P = b0 +b1 T + e versus fitted values will be useful here.) 1.2.2 Draw a scatter plot of log(P) versus T, and compare to the plot in the last problem. Is this scatter plot more nearly described by a straight line? 1.2.3 Fit the simple regression model for log(P) on T; that is, fit the model log(P)=b0 + b1 T + e and compute the relevant summary statistics (estimates of parameters, tests, analysis of variance table, R2). Draw the fitted line onto the plot in problem 1.2.2. Obtain the residual plot (versus fitted values) and compare to that fit in 1.2.1. 1.2.4 Obtain 95% confidence intervals for b0 and b1. 1.2.5 Obtain 90% prediction intervals for log(P) for predictions at 185 and at 212¡F. 1.2.6 Qualitatively compare the results of this analysis to the results in the test for Forbes' data. That is, compare the fitted lines, estimates of residual variability, prediction intervals, etc. What do you conclude? In Chapter 7 we will learn tests for comparing regression in different groups. Source: Weisberg, S. Applied Linear Regression, 2nd Edn. Wiley, New York, 1985.