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On page 157, we indicated how sampling distributions of differences, in particular differences of means, can be obtained. Using the same idea, we could arrive at the sampling distribution of differences of variances, say, S2 S2. It turns out, however, that this sampling distribution is rather complicated. Instead, we may consider 1 2 the statistic S2 >S2, since a large or small ratio would indicate a large difference while a ratio nearly equal to 1 1 2 would indicate a small difference. Theorem 5-8 Let two independent random samples of sizes m and n, respectively, be drawn from two normal populations with variances s2, s2, respectively. Then if the variances of the random sam1 2 ples are given by S2, S2, respectively, the statistic 1 2 mS2 >(m 1 nS2 >(n 2 1, n 1)s2 1 1)s2 2 S2 >s2 1 1 S2 >s2 2 2

The frequency response of a network is the value of the network function H s at s j!. This observation can be used to evaluate H j! graphically. The graphical method can produce a quick sketch of H j! and bring to our attention its behavior near a pole or a zero without the need for a complete solution.

(23)

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As shown in the tables above, there are a number of blank rows and blank columns. Requirement 5 is related to vacancies. There is not explicit handling of vacancies, although a vacancy should be the absence of a reservation on a particular date. Requirement 6.3 is the customer phone number, and it is missing. Requirement 6.5 is the agreed upon price, which is missing from the reservation information. Requirement 7.1 mentions the 1 day s payment, which is also not in the attributes. Column A.1 is the daylist, which is included to help search for vacancies. B and B.1 are necessary but not specific to a requirement. C.8 is a constructor. D.1 through D.4 are details of the transactions, which are neglected in the requirements.

EXAMPLE 12.4 Find poles and zeros of H s 10s= s2 2s 26 . Place them in the s-domain and use the polezero plot to sketch H j! . H s has a zero at z1 0. Its poles p1 and p2 are found from s2 2s 26 0 so that p1 1 j5 and p2 1 j5. The pole-zero plot is shown in Fig. 12-14(a). The network function can then be written as H s 10 s z1 s p1 s p2

Many other statistics besides the mean and variance or standard deviation can be defined for samples. Examples are the median, mode, moments, skewness, and kurtosis. Their definitions are analogous to those given for populations in 3. Sampling distributions for these statistics, or at least their means and standard deviations (standard errors), can often be found. Some of these, together with ones already given, are shown in Table 5-1.

Remarks This is true for large or small samples where the population is infinite or sampling is with replacement. The sampling distribution of means is very nearly normal (asymptotically normal) for n 30 even when the population is nonnormal. mX m, the population mean in all cases.

For each value of s, the term s z1 is a vector originating from the zero z1 and ending at point s in the s-domain. Similarly, s p1 and s p2 are vectors drawn from poles p1 and p2 , respectively, to the point s. Therefore, for any value of s, the network function may be expressed in terms of three vectors A, B, and C as follows: H s 10 A B C where A s z1 ; B s p1 , and C s p2

For n 30, the sampling distribution of the medians is very nearly normal. The given result holds only if the population is normal or approximately normal. mmed m

The magnitude and phase angle of H s at any point on the s-plane may be found from: jH s j 10 jAj jBj jCj H s A B C 5a 5b

For n 100, the sampling distribution of S is very nearly normal. sS is given by (1) only if the population is normal (or approximately normal). If the population is nonnormal, (2) can be used. Note that (2) reduces to (1) when m4 3s4, which is true for normal populations. For n 100, mS s very nearly. The remarks made for standard deviations apply here as well. Note that (2) yields (1) in case the population is normal.

1)s2 >n 30).

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