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The Wilcoxon signed-rank test can be applied to paired data. Let (X1j, X2j), j 1, 2, . . . , n be a collection of paired observations from two continuous distributions that differ only with respect to their means. (It is not necessary that the distributions of X1 and X2 be symmetric.) This assures that the distribution of the differences Dj X1j X2j is continuous and symmetric. Thus, the null hypothesis is H0: 1 0. We initially con2, which is equivalent to H0: D sider the two-sided alternative H1: 1 (or H1: D 0). 2 To use the Wilcoxon signed-rank test, the differences are rst ranked in ascending order of their absolute values, and then the ranks are given the signs of the differences. Ties are assigned average ranks. Let W be the sum of the positive ranks and W be the absolute value of the sum of the negative ranks, and W min(W , W ). If the observed value w w*, the null hypothesis H0: 1 0) is rejected where w* is chosen from Appendix Table VIII. 2 (or H0: D For one-sided tests, if the alternative is H1: 1 0), reject H0 if w w*; 2 (or H1: D and if H1: 1 (or H1: D 0), reject H0 if w w*. Be sure to use the one-sided test 2 signi cance levels shown in Appendix Table VIII.

EXAMPLE 15-5

EXAMPLE 37-6

We will apply the Wilcoxon signed-rank test to the fuel-metering device test data used previously in Example 15-3. The eight-step hypothesis-testing procedure can be applied as follows: 1. The parameters of interest are the mean fuel mileage performance for the two metering devices.

(continued)

2. H0: 1 2 or, equivalently, H0: 3. H1: 1 2 or, equivalently, H1: 4. 0.05 5. The test statistic is w

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var aMatrix:Array = [2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0]; a.filters = [new ColorMatrixFilter(aMatrix)]; //brighten method 2: offset var bMatrix:Array = [1, 0, 0, 0, 60, 0, 1, 0, 0, 60, 0, 0, 1, 0, 60, 0, 0, 0, 1, 0]; b.filters = [new ColorMatrixFilter(bMatrix)]; } } } import flash.display.Loader; import flash.net.URLRequest; import flash.system.LoaderContext; class TestImage extends Loader { public function TestImage(scale:Number = 1, x:Number = 0, y:Number = 0) { //photo (CC-BY) Roger Braunstein //source http://www.flickr.com/photos/rogerimp/2940373537/ var url:String = "http://actionscriptbible.com/files/heiwadoori.jpg"; load(new URLRequest(url), new LoaderContext(true)); scaleX = scaleY = scale; this.x = x; this.y = y; } }

where w and w are the sums of the positive and negative ranks of the differences in Table 15-2. 6. Since 0.05 and n 12, Appendix Table VIII gives the critical value as w* 13. 0.05 We will reject H0: D 0 if w 13. 7. Computations: Using the data in Table 15-2, we compute the following signed ranks:

You can use a color matrix to tint an image. There are two ways to apply the tint, and they correspond to the two ways of tinting via a ColorTransform object: multipliers and offsets. Using a ColorMatrixFilter object, the multipliers are the diagonal elements represented by r, g, b, and a in the following matrix.

Car 7 12 8 6 2 4 5 1 9 10 11 3 Difference 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.8 0.9 1.0 1.1 1.3 Signed Rank 1 2 3 4 5 6.5 6.5 8 9 10 11 12

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22.5. Therefore, min155.5, 22.52 22.5

The range that is typically useful for the multipliers is from 1 to 1. Changing the multipliers in a matrix that otherwise looks like an identity matrix is called scaling the color because the effect of such a matrix is that it simply multiplies one or more of the color components (red, blue, green, or alpha).

Conclusions: Since w 22.5 is not less than or equal to w* 13, we cannot reject the 0.05 null hypothesis that the two metering devices produce the same mileage performance.

In this matrix, r, g, b, and a represent the red, green, blue, and alpha offsets. The range that is typically useful for offsets is from 255 to 255. Example 37-7 compares an unmodi ed original image, one tinted by scaling down the green and blue channels to emphasize the red channel, and one tinted by topping out the red channel with an offset. EXAMPLE 37-7

15-3.4

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