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while mX is still given by (4). Note that (6) reduces to (5) as N S `. Theorem 5-4 If the population from which samples are taken is normally distributed with mean m and variance s2, then the sample mean is normally distributed with mean m and variance s2>n. Theorem 5-5 Suppose that the population from which samples are taken has a probability distribution with mean m and variance s2 that is not necessarily a normal distribution. Then the standardized vari# able associated with X, given by # X m s>!n

p = 1000; r = .06; a = p(1 + r / 4)4 t ; b = p(1 + r / 12)12 t ; c = p(1 + r / 365)365 t ; d = p Exp[r t];

Z is asymptotically normal, i.e., lim P(Z nS` z)

One could also argue that the modest performance requirements of this application mean that we should use the serializeable isolation level Such precaution will give us the most protection, and no one will notice any performance impairment, since the application is undemanding of high performance..

Theorem 5-5 is a consequence of the central limit theorem, page 112. It is assumed here that the population is infinite or that sampling is with replacement. Otherwise, the above is correct if we replace s> !n in (7) by sX as given by (6).

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tt = PaddedForm[t, 2]; aa = PaddedForm[a, {7, 2}]; bb = PaddedForm[b, {7, 2}]; cc = PaddedForm[c, {7, 2}]; dd = PaddedForm[d, {7, 2}];

Suppose that a population is infinite and binomially distributed, with p and q 1 p being the respective probabilities that any given member exhibits or does not exhibit a certain property. For example, the population may 1 be all possible tosses of a fair coin, in which the probability of the event heads is p 2. Consider all possible samples of size n drawn from this population, and for each sample determine the statistic that is the proportion P of successes. In the case of the coin, P would be the proportion of heads turning up in n tosses. Then we obtain a sampling distribution of proportions whose mean p and standard deviation p are given by mp p sp pq An p(1 A n p) (9)

list = Table[{tt, aa, bb, cc, dd},{t, 1, 10}]; TableForm[list, TableHeadings {None, {"year"," quarterly"," year 1 2 3 4 5 6 7 8 9 10 quarterly 1061.36 1126.49 1195.62 1268.99 1346.86 1429.50 1517.22 1610.32 1709.14 1814.02 monthly"," daily","continuously"}}] continuously 1061.84 1127.50 1197.22 1271.25 1349.86 1433.33 1521.96 1616.07 1716.01 1822.12

!pq. which can be obtained from (4) and (5) on placing m p, s For large values of n (n 30), the sampling distribution is very nearly a normal distribution, as is seen from Theorem 5-5. For finite populations in which sampling is without replacement, the second equation in (9) is replaced by sx as given by (6) with s !pq.

Note that equations (9) are obtained most easily on dividing by n the mean and standard deviation (np and !npq) of the binomial distribution.

{{a, b, c}, {d, e, f}, {g, h, i}}

Suppose that we are given two populations. For each sample of size n1 drawn from the first population, let us compute a statistic S1. This yields a sampling distribution for S1 whose mean and standard deviation we denote by mS1 and sS1, respectively. Similarly for each sample of size n2 drawn from the second population, let us compute a statistic S2 whose mean and standard deviation are mS2 and sS2, respectively. Taking all possible combinations of these samples from the two populations, we can obtain a distribution of the differences, S1 S2, which is called the sampling distribution of differences of the statistics. The mean and standard deviation of this sampling distribution, denoted respectively by mS1 S2 and sS1 S2, are given by mS1

SOCIAL ISSUES 91 Should software be copyrightable or patentable Ignoring the law for the moment, argue the question from the Kantian, Utilitarian, and Social Contract Theory perspectives While there is not a single correct answer to this question for all people, answers to this question should include observations along the following lines: The Kantian would ask the question, Am I comfortable with everyone being able to copyright or patent their software The Utilitarian would try to sum the advantages and disadvantages to all concerned The writers would gain because they could demand payment for use of their software The users would lose because buying software would be more expensive On the other hand users might gain because more software may become available to buy.

{{aa, bb, cc}, {dd, ee, ff}, {gg, hh, ii}}

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