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The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determining this number for a biodiesel fuel is expensive and time consuming. The article ?Relating the Cetane Number of Biodiesel Fuels to Their Fatty Acid Composition: A Critical Study? (J. of Automobile Engr., 2009: 565–583) included the following data on x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100 g of oil. x: 132.0 129.0 120.0 113.2 105.0 92.0 84.0 y: 46.0 48.0 51.0 52.1 54.0 52.0 59.0 x: 83.2 88.4 59.0 80.0 81.5 71.0 69.2 y: 58.7 61.6 64.0 61.4 54.6 58.8 58.0 The necessary summary quantities for hand calculation can be obtained by placing the x values in a column and the y values in another column and then creating columns for x², xy, and y² (the latter value is not needed at the moment but will be used shortly). Calculating the column sums gives ?xi = 1307.5, ?yi = 779.2, ?xi² = 128,913.93, ?xiyi = 71,347.30, ?yi² = 43,745.22 from which x? = 1307.5 / 14 = 93.392857, y? = 779.2 / 14 = 55.657143 Sxx = 128,913.93 – (1307.5)²/14 = 6802.7693 Sxy = 71,347.30 – (1307.5)(779.2)/14 = –1424.41429 Thus b = –1424.41429 / 6802.7693 = –.20938742 a = 55.657143 – (–.20938742)(93.392857) = 75.212432 and the equation of the least squares line is ? = 75.212 – .2094x, exactly that reported in the cited article. Figure 3.11 generated by the statistical computer package Minitab shows that the least squares line is a very good summary of the relationship between the two variables. A prediction of the cetane number when the iodine value is 100 is ? = 75.212 – .2094(100) = 54.27. The slope of the least squares line tells us that a decrease of roughly .209 in cetane number is associated with a 1-gram increase in iodine value.

          The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determining this number for a biodiesel fuel is expensive and time consuming. The article ?Relating the Cetane Number of Biodiesel Fuels to Their Fatty Acid Composition: A Critical Study? (J. of Automobile Engr., 2009: 565–583) included the following data on x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100 g of oil. x: 132.0 129.0 120.0 113.2 105.0 92.0 84.0 y: 46.0 48.0 51.0 52.1 54.0 52.0 59.0 x: 83.2 88.4 59.0 80.0 81.5 71.0 69.2 y: 58.7 61.6 64.0 61.4 54.6 58.8 58.0 The necessary summary quantities for hand calculation can be obtained by placing the x values in a column and the y values in another column and then creating columns for x², xy, and y² (the latter value is not needed at the moment but will be used shortly). Calculating the column sums gives ?xi = 1307.5, ?yi = 779.2, ?xi² = 128,913.93, ?xiyi = 71,347.30, ?yi² = 43,745.22 from which x? = 1307.5 / 14 = 93.392857, y? = 779.2 / 14 = 55.657143 Sxx = 128,913.93 – (1307.5)²/14 = 6802.7693 Sxy = 71,347.30 – (1307.5)(779.2)/14 = –1424.41429 Thus b = –1424.41429 / 6802.7693 = –.20938742 a = 55.657143 – (–.20938742)(93.392857) = 75.212432 and the equation of the least squares line is ? = 75.212 – .2094x, exactly that reported in the cited article. Figure 3.11 generated by the statistical computer package Minitab shows that the least squares line is a very good summary of the relationship between the two variables. A prediction of the cetane number when the iodine value is 100 is ? = 75.212 – .2094(100) = 54.27. The slope of the least squares line tells us that a decrease of roughly .209 in cetane number is associated with a 1-gram increase in iodine value.

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determining this number for a biodiesel fuel is expensive and time consuming. The article ?Relating the Cetane Number of Biodiesel Fuels to Their Fatty Acid Composition: A Critical Study? (J. of Automobile Engr., 2009: 565–583) included the following data on x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100 g of oil. x: 132.0 129.0 120.0 113.2 105.0 92.0 84.0 y: 46.0 48.0 51.0 52.1 54.0 52.0 59.0 x: 83.2 88.4 59.0 80.0 81.5 71.0 69.2 y: 58.7 61.6 64.0 61.4 54.6 58.8 58.0 The necessary summary quantities for hand calculation can be obtained by placing the x values in a column and the y values in another column and then creating columns for x², xy, and y² (the latter value is not needed at the moment but will be used shortly). Calculating the column sums gives ?xi = 1307.5, ?yi = 779.2, ?xi² = 128,913.93, ?xiyi = 71,347.30, ?yi² = 43,745.22 from which x? = 1307.5 / 14 = 93.392857, y? = 779.2 / 14 = 55.657143 Sxx = 128,913.93 – (1307.5)²/14 = 6802.7693 Sxy = 71,347.30 – (1307.5)(779.2)/14 = –1424.41429 Thus b = –1424.41429 / 6802.7693 = –.20938742 a = 55.657143 – (–.20938742)(93.392857) = 75.212432 and the equation of the least squares line is ? = 75.212 – .2094x, exactly that reported in the cited article. Figure 3.11 generated by the statistical computer package Minitab shows that the least squares line is a very good summary of the relationship between the two variables. A prediction of the cetane number when the iodine value is 100 is ? = 75.212 – .2094(100) = 54.27. The slope of the least squares line tells us that a decrease of roughly .209 in cetane number is associated with a 1-gram increase in iodine value.

Added by Kelsey W.

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