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Unformatted text preview: “Student” was the pseudonym of
W. S. Gossett, whose employer, the
Guinness Breweries of keland, 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 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 E and s, the sample
mean and the sample standard deviation. The conﬁdence interval is an expression
stating that the true mean, ,u, is likely to lie within a certain distance fmm the mea
sured mean, E. The conﬁdence interval of ,u is given by is; [eti— Conﬁdence interval: (4—3) where s is the measured standard deviation, :1 is the number of observations, and t is
Student’s r, taken from Table 4—2. Remember that in this table the degrees of freedom
are equal to n e 1. If there are ﬁve data points, there are four degrees of freedom. Values of Student's t Conﬁdence level {079) Degrees offreedom 50 90 95 98 99 99.5 9.9. .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 1—.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 —.604 5.598 8.610 5 0.727 2.015 2.571 3.365 —.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.—57 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 3.29] Note: In calculating conﬁdence intervals, 0* may be substituted for s in Equation 43 if you have a great deal of experience with a particular method and have therefore determined its “true” population standard deviation. If a" is used instead of s. the value of r to use in Equation 43 comes from the
bottom row of Table 4—2. ' {cataraesgi ctariaiessinamzs 130
129 — " “
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. 12.7 ~
«E _ “ 50% 90%
SOLUTION First we calculate E = 12.54 and s = 0.40 for the ﬁve measurements. .2‘6 _ Chance Chance
To ﬁnd the 50% conﬁdence interval, look up I in Table 42 under 50 and across 3 _ E 8:? me > til? true
from four degrees of Freedom (degrees of freedom 2 n — 1). The value of r is 0.741, 125 _ Eagle 22:8
so the conﬁdence interval is E _ J in this in this
3 12.4 _ interval Interval
_ ts (0.740(040) 3 _
5097 = i — =12.5 i — =12.5 10.1 D
M 0) x W 4 V5 4 3 123 _
The 90% conﬁdence interval is 122 _
._ ts (2.132)(0.40) _
9097 = i — = 12.5 i — = 12.5 10.3 1211‘
M 0) x V; 4 V3 4 3 _
12.0 L These calculations mean that there is a 50% chance that the true mean, it, lies in
the range 12.54 30.13 (12.41 to 12.67). There is a 90% chance that it lies in the You mjght appreciate Box 4] at this
range 12.54 £0.33 (12.15 to 12.92). time. ...
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