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inter 90% confidence is = 12.5. VII The confidence interval many time 90 % repeat sethelude H. u Nine-ter 1 expected are If we expected to include to the true mean, measured on inter 1 20 14 TEST YOURSELF six results, wt%) confidence ,0888° Hydro Using Vo = 12.50 + 0.31 .007 91 5 4 TABLE 4-4 Values of Student's t Confidence level (%) 99 98 95 Degrees of freedom 50 90 6.314 12.706 31.821 63.656 1.000 2.920 4.303 6.965 9.925 I 2 0.816 3.182 4.541 5.841 0.765 2.353 3 0.741 2.132 2.776 3.747 4.604 4 2.015 2.571 3.365 4.032 5 0.727 6 0.718 1.943 2.447 3.143 3.707 7 0.711 1.895 2.365 2.998 3.500 8 0.706 1.860 2.306 2.896 3.355 9 0.703 1.833 2.262 2.821 3.250 10 0.700 1.812 2.228 2.764 15 3.169 0.691 1.753 2.131 20 2.602 0.687 1.725 2.947 25 2.086 0.684 2.528 1.708 30 2.060 2.845 0.683 40 1.697 2.485 2.042 2.787 0.681 1.684 2.457 60 0.679 2.021 2.750 120 1.671 2.423 0.677 2.000 2.704 80 1.658 1.980 2.390 0.674 1.645 calculating confidence 1.960 2.358 2.660 In population intervals, standard o may be 2.326 2.617 deviation. If substituted o is used for S instead in Values of 1 in this table of 5, the Equation apply value 4-7 to two-tailed of if 1 you to have 2.576 tailed use in a great deal test, we use values of I listed for 90% tests confidence. illustrated Each in Figure wing outside 4.9a. The of t Equation for 95% confidence 4-7 comes level from of the experience bottom row with of a The 90% confidence contains specifies 5% this particular table, 2.5% method of the a standard dom from a Gaussian meaning of Interval confider Figure Meaning 4-5 illustrates of a Confidence curve, their mean deviation (a) population X = Ux = k + n 1 + m2 - (y - V)2 2 = (4-27) (C (xi X) 2 15 V av where Sy is the standard deviation of y (Equation 4-20), Iml is the absolute value of the slope (= X = av ABS(m) in Excel), k is the number of replicate measurements of the unknown, n is the number of data points for the calibration line (14 in Table 4-8), y is the mean value of y for the points on the calibration line, Xi are the individual values of X for the points on the calibration line, and X is the mean value of X for the points on the calibration line. For a single measurement of the unknown, k = 1 and Equation 4-27 gives Ux = +0.39 g. Four replicate unknowns (k = 4) with an average corrected absorbance of 0.302 reduce the uncertainty to Ux = +0.23 g. The confidence interval for X is where t is Student's t (Table 4-4) for n - 2 degrees To find y of freedom. If Ux = 0.23 g and n = 14 points (12 degrees of freedom), the 95% confidence use the interval for X is ttux = +(2.179)(0.23) = +0.50 g. There is no 1/Vn in the expression for confidence interval because Ux is the standard deviation of the mean. TINV(0.0 Propagation of Uncertainty You now have all the tools required for a more rigorous discussion of propagation of uncertainty than we had in Chapter 3. If you are so inclined, you will find that discussion in Appendix B. 4-9 A Spreadsheet for Least Squares Figure 4-15 implements least-squares analysis, including propagation of error with Equation 4-27. Enter values of X and y in columns B and C. Then select cells B10:C12. Enter the for- mula "=LINEST(C4:C7,B4:B7,TRUE,TRUE)" and press CONTROL + SHIFT + ENTER on a PC or COMMAND(% + RETURN on a Mac. LINEST returns m, b, Um, Ub, R², and Sy in A B C D E 1 F Least-Squares Spreadsheet G 2 3 Highlight cells B10:C12 x y 6 4 Type "= LINEST(C4:C7, 1 2 5 B4:B7,TRUE,TRUE) y = 0.6154x + 1.3462 3 3 6 5 For PC, press 4 4 7 CTRL+SHIFT+ENTER 6 5 8 For Mac, press 4 9 COMMAND+RETURN LINEST output: 10 m 0.6154 1.3462 11 b 3 Um 0.0544 12 0.2141 ub R² 0.9846 13 0.1961 Sy 2 14 n = 4 15 B14 = COUNT(B4:B7) Mean y = 16 3.5 (x) - mean x) 2 = B15 = AVERAGE(C4:C7 1 17 13 B16 = DEVSQ(B4:B7) 18 Measured y = 2.72 Input o 1 19 k = Number of replicate 2 3 4 measurements of y = x 20 1 Input 21 Derived x = 2.2325 B20 = (B18-C10)/B10 Ux = FIGURE 4-15 Spreadsheet for linear least-squares 0.3735 B21 analysis. = (C12/ABS(B10)) SQRT ((1/B19)+(1/B14). + 4-9 Spreader fercury be many times greater than that of water. Contamination of fish by mercury is a problem is absorbed by fish when they filter water through their gills, and the concentration of mercury because in fish it an neat have can negative effects on the brain, especially in developing children. High levels of mercury can also have legative health impacts on adults. A scientist working for the environmental protection agency (EPA) is working to analyze mercury levels in L fish of believed to be contaminated. She digests 51.35 grams of tuna fish and dilutes it to a total volume of 1.000 She then uses atomic absorbance spectrophotometry to determine the concentration of the sample. She uses water. an external calibration method where she analyzes four standards containing mercury at increasing concentrations, a blank, and the unknown fish sample (one analysis on each sample) and gets the following results: Concentration of mercury Sample (ppm) absorbance Std 1 5.00 0.602 Std 2 3.00 0.380 Std 3 1.00 0.167 Std 4 0.50 0.127 blank .020 Unknown 0.297 Your job is to analyze her data using Microsoft Excel, and use it to get the information required on the next sheet. 1 - Hint: A note about significant figures. As a general rule, the uncertainty (in this case the standard deviation) should be reported with 1 or 2 significant figures. The measurement number associated with that uncertainty can then be rounded to the same decimal place (not the same # of significant figures) as the uncertainty. For example, if m (the slope of the calib curve) = 2.3456 and Sm (the uncertainty in the slope) = 0.04253, then the proper way to express the slope with uncertainty rounded to 2 significant figures would be: 2.346 0.042. It would also be acceptable to round the uncertainty to only 1 significant figure, i.e. 2.35 I 0.04. I will accept answers where the uncertainty is rounded to 1 or 2 significant figures, just be consistent and be sure that the measurement number matches the decimal of the uncertainty. It is NOT acceptable to use more than 2 significant figures in the uncertainty. 1. Slope of calibration curve (m) = 2. Uncertainty of slope (Sm) = 3. Intercept of calibration curve (b) = 4. Uncertainty of intercept (Sb) = 5. Concentration of mercury in the water sample (ppm) = 6. Uncertainty in concentration of mercury in water (Sx) (ppm)= 7. Concentration of mercury in the tuna (in mg/kg) = 8. Uncertainty in conc. of mercury in the tuna (mg/kg) = 9. 95% confidence interval for concentration of mercury in tuna (mg/kg)* = ote: To calculate the 95% confidence interval from calibration data, use the equation = tSx, where Sx is the certainty you calculated for the concentration of tuna in mg/kg, and t is for N-2 degrees of freedom (where N the number of data points in the calibration curve). In other situations we have calculated a confidence erval with the equation ts/Nn. However, the equation that calculates the uncertainty in X (equation 4-27 of rris 9th edition) already divides by Vn, therefore, it is not necessary to include this Vn term in your calculation the confidence interval when you use calibration data. Note that N-2 degrees of freedom (rather than N-1 grees of freedom) is used because you have both X and y data.

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