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be inserted backwards. The cipher and control rotors each permute the 26 letters. The five index rotors each permute the numbers 0 through 9 and, of course, the index rotors are not interchangeable with the other rotors. Unlike the cipher and control rotors, the index rotors cannot operate in the reverse orientation. Figure 2.15 illustrates the cryptographic components of Sigaba in encryption mode.

EXAMPLE 10-1

A product developer is interested in reducing the drying time of a primer paint. Two formulations of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a new drying ingredient that should reduce the drying time. From experience, it is known that the standard deviation of drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingredient. Ten specimens are painted with formulation 1, and another 10 specimens are painted with formulation 2; the 20 specimens are painted in random order. The two sample average drying times are x 1 121 minutes and x 2 112 minutes, respectively. What conclusions can the product developer draw about the effectiveness of the new ingredient, using 0.05 We apply the eight-step procedure to this problem as follows:

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Figure 2.15: Sigaba cncryption. After a. letter is encrypted or decrypted, from one to four of the cipher rotors st.cp. The nuniber and selection of the stepping cipher rotors is controlled by thc other two banks of rotors, that is, the control and index rotors. For each letter typed, the rightmost control rotor receives four simultancous inputs, which we assume to be F, G, H, and I. These four letters are perniutetl xcording to the five control rotors and the resulting four permutationdcpcrident output letters are combined before being input to the index rotor bank. Let I, denote t>heinput to element j of the leftmost index rotor and A through Z the outputs of the control rotors. Then

1. 2. 3. 4. 5.

The quantity of interest is the difference in mean drying times, 0, or H 0: 1 H 0: 1 2 2.

H 1:

11 = 12 =

182 2

64 and n1 n2 10. 1.645 z0.05. 121 minutes and x2 112 minutes, the test statistic is 121 112 182 2 182 2 10 B 10 2.52

66 66 67 68 68 69 69 71 72 73 74 75 75 77 77 81

(2.7)

Conclusion: Since z0 2.52 1.645, we reject H0: 1 0.05 level 2 at the and conclude that adding the new ingredient to the paint signi cantly reduces the drying time. Alternatively, we can nd the P-value for this test as P-value Therefore, H0:

0.0059 0.0059.

where (2.7) is interpreted to mean that, for example, input 3 of the leftmost index rotor is active if output D or E (or both) results from the control rotors;

10-3.1 Hypothesis Tests for a Difference in Means, Variances Unknown 337 10-3.2 More About the Equal Variance Assumption (CD Only) 344 10-3.3 Choice of Sample Size 344 10-3.4 Con dence Interval on a Difference in Means, Variances Unknown 345

9-3.1 Hypothesis Tests on the Mean 300 9-3.2 P-Value for a t-Test 303 9-3.3 Choice of Sample Size 304 9-3.4 Likelihood Ratio Approach to Development of Test Procedures (CD Only) 305

10-5.1 The F Distribution 355 10-5.2 Development of the F Distribution (CD Only) 357 10-5.3 Hypothesis Tests on the Ratio of Two Variances 357 10-5.4 -Error and Choice of Sample Size 359 10-5.5 Con dence Interval on the Ratio of Two Variances 359

otherwise input 3 is inactive Note that 10 is missing, which implies that input 0 is always inactive Since four values are input to the control rotors, due to the OR of the outputs, anywhere from one to four of the inputs to the index rotors are active at each step The middle three control rotors step in an odometer-like fashion-almost The fast, medium, and slow control rotors are indicated by F, hl, and S, respectively, in Figure 215, where the fast rotor steps with each letter, the mcdium rotor steps once for each 26 steps of the fast rotor, and the slow rotor steps once for each 26 steps of the medium rotor The stepping of these three rotors differs from an odometer only in the order of the fast, medium and slow rotors.

10-6

10-6.1 Large-Sample Test for H0 : p1 p2 361 10-6.2 Small Sample Test for H0 : p1 p2 (CD Only) 364 10-6.3 -Error and Choice of Sample Size 364 10-6.4 Con dence Interval for P1 P2 365

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