Statistics for Environmental Engineers

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-1.842


y- — -n = 7.51-8.00 s /Xn    1.38/727


with v = 26 degrees of freedom. Find the probability of such a value of t occurring by referring to the tabulated tail areas of the t distribution in Appendix A. Because of symmetry, this table


6


4


Sampling distribution of the variance



Sampling distribution of t



14-| 7 10 6



40 random samples of n = 4



•и . •:    *• «•    •



Parent


distribution


N(10,1)



12


y=^10-n8



Sampling distribution of the mean is normal


s2 JUy—y)22 n-1


0 2


*=0


s n


-2 -I


FIGURE 2.11 Forty random samples of n = 4 from a N(10,1) normal distribution to produce the sampling distributions of y , s2, and t. (Adapted from Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, New York, Wiley Interscience.)

P(y < 7.51) = 0.04    P(f < 1.853) = 0.04


FIGURE 2.12 The y and t reference distributions for the sample average of the nitrate data of Example 2.1.


serves for negative as well as positive values. For v = 26, the tail areas are 0.05 for t = -1.706,


0.025 for t = -2.056, and 0.01 for t = -2.479. Plotting these and drawing a smooth curve as an aid to interpolation gives Prob(t < -1.853) « 0.04, or only about 4%. This low probability suggests that there may be a problem with the measurement method in this laboratory.

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