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(a) Based on these 100 observations, is a Poisson distribution with a mean of 1.2 an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test. 9-60. Let X denote the number of flaws observed on a large coil of galvanized steel. Seventy-five coils are inspected and the following data were observed for the values of X: Values Observed Frequency 1 1 2 11 3 8 4 13 5 11 6 12 7 10 8 9

(a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this data Use 0.05. (b) Calculate the P-value for this test. 9-62. Consider the following frequency table of observations on the random variable X: Values Frequency 0 4 1 21 2 10 3 13 4 2

(a) Does the assumption of the Poisson distribution seem appropriate as a probability model for this data Use 0.01. (b) Calculate the P-value for this test. 9-61. The number of calls arriving at a switchboard from noon to 1 PM during the business days Monday through Friday is monitored for six weeks (i.e., 30 days). Let X be

In contrast, Purple uses switches, where each step of a switch changes to a different permutation That is, instead.

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(a) Based on these 50 observations, is a binomial distribution with n 6 and p 0.25 an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test.

9-63. De ne X as the number of under lled bottles from a lling operation in a carton of 24 bottles. Sixty cartons are inspected and the following observations on X are recorded: Values Frequency 0 39 1 23 2 12 3 1

Figure 2.9: Purple encryption. of using permutations wired to rotors-which only requires one permutation per rotor-each step of Purple s S , L , M , and R switches t o a different, unrelated, hardwired permutation. Consequently, we refer to S , L , M , and R as switches and Purple as a stepping switch machine. While the distinction between rotors and switches might seem relatively minor, it is actually a major difference. For one thing, a stepping switch machine like Purple is inherently more complex and difficult to engineer than a rotor machine. From a cryptanlytic point of view, a rotor machine provides an elegant way to generate a large number of permutations, while a comparable stepping switch machine must be far more complex. There are also significant differences between the types of permutations that can be generated by Purple and Enigma, as discussed below and in Problem 16. The Purple encryption formula depending on whether the letter being encrypted corresponds to a six or twenty . Let x be the given input letter and let y be the corresponding output letter. Then we can denote the encryption by

(a) Based on these 75 observations, is a binomial distribution an appropriate model Perform a goodness-of- t procedure with 0.05. (b) Calculate the P-value for this test. 9-64. The number of cars passing eastbound through the intersection of Mill and University Avenues has been tabulated by a group of civil engineering students. They have obtained the data in the adjacent table: (a) Does the assumption of a Poisson distribution seem appropriate as a probability model for this process Use 0.05. (b) Calculate the P-value for this test.

Many times, the n elements of a sample from a population may be classi ed according to two different criteria. It is then of interest to know whether the two methods of classi cation are statistically independent; for example, we may consider the population of graduating engineers, and we may wish to determine whether starting salary is independent of academic disciplines. Assume that the rst method of classi cation has r levels and that the second method has c levels. We will let Oij be the observed frequency for level i of the rst classi cation method and level j on the second classi cation method. The data would, in general, appear as shown in Table 9-2. Such a table is usually called an r c contingency table. We are interested in testing the hypothesis that the row-and-column methods of classi cation are independent. If we reject this hypothesis, we conclude there is some interaction between the two criteria of classi cation. The exact test procedures are dif cult to obtain, but an approximate test statistic is valid for large n. Let pij be the probability that a randomly selected element falls in the ijth cell, given that the two classi cations are independent. Then pij uivj,

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