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ConInterval

# ConInterval - Below is an excerpt from Harris D.C Exploring...

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Unformatted text preview: Below is an excerpt from Harris, D.C., Exploring Chemical Analysisfreeman, New York, 1997, 58 concerning conﬁdence intervals and how to calculate them. “Student” was the pseudonym of W. S. Gussett, whose employer, the Guinness Breweries of Ireland, re— stricted publications for proprietary reasons. Because of the importance of his work, Gossett published it under an assumed name in 1908. Student’s t is the statistical tool used to express conﬁdence intervals and to com- pare results from different experiments. You can use it to evaluate the probability that your red blood cell count will be found in a certain range on “normal” days. Confidence Intervals From a limited number of measurements, it is impossible to ﬁnd the true mean, ,u, or the true standard deviation, 0-. What we can determine are i and s, the sample mean and the sample standard deviation. The conﬁdence interval is an expression stating that the true mean, ,u, is lilceiy to lie within a certain distance from the mea- sured mean, 3. The conﬁdence interval of it is given by Conﬁdence interval: (4—3) where s is the measured standard deviation, n is the number of observations, and r is Student’s it, taken from Table 4—2. Remember that in this table the degrees of ﬁeedom are equal to n — 1. If there are ﬁve data points, there are four degrees of freedom. Values of Student's t Conﬁdence level {96) Degrees of freedom 50 90 95 98 99 99.5 99.9 1 1.000 6.314 12.706 31.821 63.657 127.32 636.619 2 0.816 2.920 4.303 6.965 9.925 14.089 31.598 3 0.765 2.353 3.182 4.541 5.841 7.453 12.924 4 0.741 2.132 2.776 3.747 4.604 5.598 8.610 5 0.727 2.015 2.571 3.365 4.032 4.773 6.869 6 0.718 1.943 2.447 3.143 3.707 4.317 5.959 7 0.711 1.895 2.365 2.998 3.500 4.029 5.408 8 0.706 1.860 2.306 2.896 3.355 3.832 5.041 9 0.703 1.833 2.262 2.821 3.250 3.690 4.781 10 0.700 1.812 2.228 2.764 3.169 3.581 4.587 15 . 0.691 1.753 2.131 2.602 2.947 3.252 4.073 20 0.687 1.725 2.086 2.528 2.845 3.153 3.850 25 0.684 1.708 2.068 2.485 2.787 3.078 3.725 30 0.683 1.697 2.042 2.457 2.750 3.030 3.646 40 0.681 1.684 2.021 2.423 2.704 2.971 3.551 60 0.679 1.671 2.000 2.390 2.660 2.915 3.460 120 0.677 1.658 1.980 2.358 2.617 2.860 3.373 00 0.674 1.645 1.960 2.326 2.576 2.807 3.291 Note: In calculating conﬁdence intervals, or may be substituted for s in Equation 4-3 if you have a great deal of experience with a particular method and have therefore determined its “true” population standard deviation. If U is used instead of s, the value of I to use in Equation 4-3 comes from the bottom row of Table 4—2. Calculating Confidence intervals 13-"0 12.9 In replicate analyses, the carbohydrate content of a glycoprotein (a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohy- 123 _ drate per 100 g of protein. Find the 50 and 90% conﬁdence intervals for the car- _ bohydrate content. 12;; _ E _ 50% 90% SOLUTION First we calculate E = 12.54 and s = 0.40 for the ﬁve measurements. E 12.6 _ Chance Chance To ﬁnd the 50% conﬁdence interval, look up t in Table 4-2 under 50 and across 8 _ that true that true from four degrees of freedom (degrees of freedom 2 a — 1). The value of t is 0.741, *3 12.5 _ 8:19.116 > \$15118 so the conﬁdence interval is E. _ in this in this 3 124 _ interval interval 23 Jmore) = r : ‘—S = 12.54 i W = 12.54 :013 U - n 5 12.3 — The 90% conﬁdence interval is 122 J[709096) =5 i % = 12.54 i W = 12.54 :033 12.1 12.0 These calculatious mean that there is a 50% chance that the true mean, pl, lies in the range 12.54 \$0.13 (12.41 to 12.67). There is a 90% chance that pt lies in the You mjght appreciate Box 4-1 at this range 12.5.; i038 (12.15 to 12.92). time. ...
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