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We can take random samples from the population and then use these samples to obtain values that serve to estimate and test hypotheses about the population parameters. By way of illustration, let us consider Example 5.1 where X is a random variable whose values are the various heights. To obtain a sample of size 100, we must first choose one individual at random from the population. This individual can have any one value, say, x1, of the various possible heights, and we can call x1 the value of a random variable X1, where the subscript 1 is used since it corresponds to the first individual chosen. Similarly, we choose the second individual for the sample, who can have any one of the values x2 of the possible heights, and x2 can be taken as the value of a random variable X2. We can continue this process up to X100 since the sample size is 100. For simplicity let us assume that the sampling is with replacement so that the same individual could conceivably be chosen more than once. In this case, since the sample size is much smaller than the population size, sampling without replacement would give practically the same results as sampling with replacement. In the general case a sample of size n would be described by the values x1, x2, . . . , xn of the random variables X1, X2, . . . , Xn. In the case of sampling with replacement, X1, X2, . . . , Xn would be independent, identically distributed random variables having probability distribution f(x). Their joint distribution would then be P(X1 x1, X2 x2, c, Xn xn) f (x1) f (x2) c f (xn) (1)

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To dispose of the Student table, one can use this command: DROP TABLE Student; When one must modify a database object like a table, the command to use is ALTER For instance, to add a birthdate column to the student table, one could use this command to add a column named Birthdate of data type Date: ALTER TABLE Student ADD COLUMN Birthdate Date; In addition to adding columns, one can use the ALTER TABLE command to drop columns, add or drop constraints, and set or drop defaults DML DATA MANIPULATION LANGUAGE The first DML statement to learn is SELECT The SELECT statement provides the means of retrieving information from the database It is a very flexible command with numberless variations and much to know about using it.

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Any quantity obtained from a sample for the purpose of estimating a population parameter is called a sample statistic, or briefly statistic. Mathematically, a sample statistic for a sample of size n can be defined as a function of the random variables X1, . . . , Xn, i.e., g(X1, . . . , Xn ). The function g(X1, . . . , Xn ) is another random variable,

p[x_] = x5 3 x4 + 2 x3 7 x + 12; list = Table[{x, p[x]},{x, 3, 3, .5}];

Random experiments, sample spaces, and events 1.1. A card is drawn at random from an ordinary deck of 52 playing cards. Describe the sample space if consideration of suits (a) is not, (b) is, taken into account.

TableForm[list] 3. 2.5 2. 1.5 1. 0.5 0. 0.5 1. 1.5 2. 2.5 3. 507. 216.594 70. 7.03125 13. 15.0313 12. 8.59375 5. 0.65625 2. 6.21875 45.

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(a) If we do not take into account the suits, the sample space consists of ace, two, . . . , ten, jack, queen, king, and it can be indicated as {1, 2, . . . , 13}. (b) If we do take into account the suits, the sample space consists of ace of hearts, spades, diamonds, and clubs; . . . ; king of hearts, spades, diamonds, and clubs. Denoting hearts, spades, diamonds, and clubs, respectively, by 1, 2, 3, 4, for example, we can indicate a jack of spades by (11, 2). The sample space then consists of the 52 points shown in Fig. 1-5.

Fig. 1-5

PaddedForm[TableForm[list],{10, 6}] 3.000000 2.500000 2.000000 1.500000 1.000000 0.500000 0.000000 0.500000 1.000000 1.500000 2.000000 2.500000 3.000000 507.000000 216.594000 70.000000 7.031000 13.000000 15.031000 12.000000 8.594000 5.000000 0.656000 2.000000 6.219000 45.000000

1.2. Referring to the experiment of Problem 1.1, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the events (a) A < B, (b) A > B, (c) A < Br, (d) Ar < Br, (e) A B, (f ) Ar Br, (g) (A > B) < (A > Br).

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