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This equation can be used to predict pull strength for pairs of values of the regressor variables wire length (x1) and die height (x2). This is essentially the same regression model given in Equation 1-7, Section 1-3. Figure 1-14 shows a three-dimentional plot of the plane of predicted values y generated from this equation.

x = 13

(3.24)

hypothetical population, because it does not physically exist. Sometimes there is an actual physical population, such as a lot of silicon wafers produced in a semiconductor factory. In previous chapters we have introduced the mean of a probability distribution, denoted . If we think of a probability distribution as a model for the population, one way to think of the mean is as the average of all the measurements in the population. For a nite population with N measurements, the mean is

(6-2)

A consequence of (3.24) is that if we know one complete Xi, we can compute all bits of X j , for j > i, assuming that the corresponding plaintexts p j are known. By using the CRC inversion formula, we can also use this Xi to recover Xj for j < i, again, provided that the corresponding plaintexts p j are known.

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The sample mean, x, is a reasonable estimate of the population mean, . Therefore, the engineer designing the connector using a 3 32-inch wall thickness would conclude, on the basis of the data, that an estimate of the mean pull-off force is 13.0 pounds. Although the sample mean is useful, it does not convey all of the information about a sample of data. The variability or scatter in the data may be described by the sample variance or the sample standard deviation. De nition If x1, x2, p , xn is a sample of n observations, the sample variance is a 1xi

x2 2 1 (6-3)

If you use Auto Reset, both le and folder numbering restarts when you insert another memory card If you prefer to organize images by memory card, then Auto Reset is a useful option Folder numbers begin with 100, and le numbering begins with 0001, unless the card contains a previous image, in which case numbering begins with the highest le number in the highest numbered folder on the memory card However, multiple images on your computer will have the same lename but will be stored in different folders Because of this, you should create separate folders on the computer and other devices for each download and otherwise follow careful folder organization to avoid lename con icts and the potential overwriting of special images on the computer.

The units of measurements for the sample variance are the square of the original units of the variable. Thus, if x is measured in pounds, the units for the sample variance are (pounds)2. The standard deviation has the desirable property of measuring variability in the original units of the variable of interest, x. How Does the Sample Variance Measure Variability To see how the sample variance measures dispersion or variability, refer to Fig. 6-2, which shows the deviations xi x for the connector pull-off force data. The greater the amount of variability in the pull-off force data, the larger in absolute magnitude some of the deviations xi x will be. Since the deviations xi x always sum to zero, we must use a measure of variability that changes the negative deviations to nonnegative quantities. Squaring the deviations is the approach used in the sample variance. Consequently, if s2 is small, there is relatively little variability in the data, but if s2 is large, the variability is relatively large.

(xz)O23 = Xi+i @ CRCtable[(Xi)24...31 @ p i + 1 ] ...

EXAMPLE 6-2

Table 6-1 displays the quantities needed for calculating the sample variance and sample standard deviation for the pull-off force data. These data are plotted in Fig. 6-2. The numerator of s2 is a 1xi

2 3 = Xi+2 @

i 1 8

x2 2

Computation of s2 The computation of s2 requires calculation of x, n subtractions, and n squaring and adding operations. If the original observations or the deviations xi x are not integers, the deviations xi x may be tedious to work with, and several decimals may have to be carried to ensure

CR,Ctable[(Xi+l)24 31 @ pi+2] ... CRCtable[(Xi+z)z4...31 @ p i + 3 ] .

Table 6-1 Calculation of Terms for the Sample Variance and Sample Standard Deviation i 1 2 3 4 5 6 7 8

12.6 12.9 13.4 12.3 13.6 13.5 12.6 13.1 104.0

(3.25) (3.26) (3.27)

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