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Randomized block designs.pdf

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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 = ˆ (ˆ) where is a quantity to be estimated (e.1. 2), ˆ is an estimate of (e.g. ¯1 ¯2) and (ˆ) is an estimate of the standard deviation of ˆ 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 = à 2 2 ! ˆ = 2 2 ³ (ˆ) ´ = ˆ · (ˆ) ˆ+ · (ˆ) 2 2 33 after a rearrangement. Thus, before we take the sample, we know that the random interval h i = ˆ 2· (ˆ) ˆ+ 2 · (ˆ) will, with probability 1, contain the true value of . After the sample is taken, and ˆ (ˆ) cal- culated numerically, we call a 100(1 )% condence interval. • In the mortar example, the di erence in averages was ˆ = 16 764 17 922 = 1 158; then from the computer output, 1 158 1 158 9 1094 = = (ˆ) (ˆ) = 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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