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Lecture

# Randomized block designs.pdf

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School
Department
Statistics
Course
STAT368
Professor
Douglas Wiens
Semester
Winter

Description
32 4. Randomized block designs  Condence intervals. The following treatment seems general enough to cover the cases of prac- tical interestAny -ratio can be written in the form =  () where is a quantity to be estimated (e.1. 2),  is an estimate of (e.g. ¯1 ¯2) and () is an estimate of the standard deviation of  r 1 1 (e.g. + ). Then if ± 2are the points 1 2 on the horizontal axis which each have 2 of the probability of the -distribution lying beyond them, we have ³ ´ 1 = Ã 2 2 !  = 2 2 ³ () ´ =  · () + · () 2 2 33 after a rearrangement. Thus, before we take the sample, we know that the random interval h i =  2· () + 2 · () will, with probability 1, contain the true value of . After the sample is taken, and  () cal- culated numerically, we call a 100(1 )% condence interval.  In the mortar example, the di erence in averages was  = 16 764 17 922 = 1 158; then from the computer output, 1 158 1 158 9 1094 = = () () = 9 1094 = 1271 There were 18 d.f., and so for a 95% interval we nd 18 2on R: ( 975 18) [1] 2 100922 Thus = 1 158 ± 2 1009 · 1271 = 1 158 ± 2670 = [ 1 4250 8910], in agreement with the computer output. 34 Sample size calculations. The power of a test is the probability of rejecting the null hypothesis when it is false.Suppose that we would like a power of at least .99 when the mortar means di er by = = 5. Furthermore, suppose that we will reject 0 if the p-value is 05. (We use = 05.) We need an estimate of , here Ill use = 4, which is somewhat larger than was. > power.t.test(delta = .5, sd = .4, sig.level = .05,power = .99,type = "two.sample", alternative = "two.sided") Two-sample t test power calculation n = 24.52528 delta = 0.5 sd = 0.4 sig.level = 0.05 power = 0.99 alternative = two.sided NOTE: n is number in *each* group Thus in a future study, if = 4 is accurate, we will need two equal samples of at least 25 each in order to get the required power. Use help(power.t.test) to get more details on this function. 35  Paired comparisons. Su
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