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1 2 4 4 3 5 5 4 2

2 1 2 2

(10-31)

2 1 7 6 5 6 0 2 1

where f 2,n2 1,n1 1 and f1 2,n2 1,n1 1 are the upper and lower 2 percentage points of the F distribution with n2 1 numerator and n1 1 denominator degrees of freedom, respectively. A con dence interval on the ratio of the standard deviations can be obtained by taking square roots in Equation 10-31.

EXAMPLE 10-13

A company manufactures impellers for use in jet-turbine engines. One of the operations involves grinding a particular surface nish on a titanium alloy component. Two different grinding processes can be used, and both processes can produce parts at identical mean surface roughness. The manufacturing engineer would like to select the process having the least variability in surface roughness. A random sample of n1 11 parts from the rst process results in a sample standard deviation s1 5.1 microinches, and a random sample of n2 16 parts from the second process results in a sample standard deviation of s2 4.7 microinches. We will nd a 90% con dence interval on the ratio of the two standard deviations, 1 2. Assuming that the two processes are independent and that surface roughness is normally distributed, we can use Equation 10-31 as follows: s2 1 f s2 0.95,15,10 2

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3 7 0 5 0 2 6 0 5

2 1 2 2 2 1 2 2

15.12 2 0.39 14.72 2

4 5 6 7 6 0 4 3 1 6 5 2 2 0 1 3 6 4 7 2 1 0 7 4 7 1 3 4 6 5 3 1 7 3 0 4

The Set-up menu is where you control the hardware and the operational and administrative aspects of the PowerShot G11. The options selected in this menu don t affect

s2 1 f s 2 0.05,15,10 2

15.12 2 2.85 14.72 2

or upon completing the implied calculations and taking square roots, 0.678

the permutation A followed by the permutation P. Find the diffwence sequence (as discussed in Section 2.3.3) for the pair PA, Q A and also for thc pair P B , Q B . b. Explain the results in part a. c. Why are these results relevant to the diagnosis of Purple 18. The permutations in Table 2.6 were generated from switched permutations using a niethod analogous to that used in the Purple cipher. Recover the L , M and R permutations (or an equivalent set of permutations). Hint: There are three permutations per switch. Use the permutations in lines 3, 4 and 5 to solve for the L permutations, lines 4, I; and 9 to solve for the M permutations, and lines 0, 10 and 11 to solve for the R permutations. 19. Consider the putative matched plaintext and ciphertext pairs in Table 2.5. Explain why these could not have resulted from a 3-legged depth of either the Enigma or Purple ciphers.

1 f0.05,10,15 1 2.54 0.39. Notice that we have used Equation 10-28 to nd f0.95,15,10 Since this con dence interval includes unity, we cannot claim that the standard deviations of surface roughness for the two processes are different at the 90% level of con dence. EXERCISES FOR SECTION 10-5

10-45. For an F distribution, nd the following: (a) f0.25,5,10 (b) f0.10,24,9 (c) f0.05,8,15 (d) f0.75,5,10 (e) f0.90,24,9 (f ) f0.95,8,15 10-46. For an F distribution, nd the following: (a) f0.25,7,15 (b) f0.10,10,12 (c) f0.01,20,10 (d) f0.75,7,15 (e) f0.90,10,12 (f) f0.99,20,10 10-47. Two chemical companies can supply a raw material. The concentration of a particular element in this material is important. The mean concentration for both suppliers is the same, but we suspect that the variability in concentration may

20. Suppose that a permutation P is wired to a rotor. Define n ( P )to be the resulting permutation when the rotor is shifted one step. For example, if P maps 0123456 to 6504213, then a ( P ) maps 0123456 to 4061532, n z ( P )maps 0123456 to 3510264, and so on (see Section 2.2.3 and Problem 1). Consider the Maroon cipher that we invented in Section 2.3. Let PL, PM and PK be given permutations of the 26 letters. Suppose that the permutations on switch L are selected to be

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