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and is said to follow the F distribution with u degrees of freedom in the numerator and v degrees of freedom in the denominator. It is usually abbreviated as Fu,v.

22 , 42

Listing 15.1 (continued)

Two F distributions are shown in Fig. 10-4. The F random variable is nonnegative, and the distribution is skewed to the right. The F distribution looks very similar to the chi-square distribution; however, the two parameters u and v provide extra exibility regarding shape. The percentage points of the F distribution are given in Table V of the Appendix. Let f ,u,v be the percentage point of the F distribution, with numerator degrees of freedom u and denominator degrees of freedom v such that the probability that the random variable F exceeds this value is P1F f

This is illustrated in Fig. 10-5. For example, if u Appendix that P1F f0.05,5,10 2

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The methods that have been defined in Listing 15.1 are shared (static), which means that it is not necessary to create an instance of the Db class first. The method can be called by using simply the name of the class as follows:

P1F5,10

f ( x)

x = Db.ExecuteScaler(cn, Select count(*) from customers )

That is, the upper 5 percentage point of F5,10 is f0.05,5,10 3.33. Table V contains only upper-tail percentage points (for selected values of f ,u,v for 0.25) of the F distribution. The lower-tail percentage points f1 ,u,v can be found as follows.

(10-28)

For example, to nd the lower-tail percentage point f0.95,5,10, note that f0.95, 5,10 1 f0.05,10, 5 1 4.74 0.211

The Db class contains a method called ExecuteScaler that is overloaded so it can be called with a connection and SQL, or with additional arguments. The ExecuteScaler method returns the first column of the first row from the result set of the query. This method is typically used to retrieve a numeric value, so the return type is cast to a Long, using the CType function. The ExecuteDataSet method contains the same overloads as the ExecuteScaler method. This method returns a DataSet that could contain many tables if the SQL command contained many select statements. The AddParameters method is a private helper method that is used to enumerate a parameter array and add each parameter to the command s parameters collection. After this code is added to the project, the project must be built. Building the project creates a .dll file, which is an assembly. The assembly can be revealed by clicking Show All Files in the Solution Explorer.

Right-click a blank area of the desktop and click New and then Shortcut. The Create Shortcut dialog box appears.

10-5.2 10-5.3

Let X11, X12, p , X1n1 be a random sample from a normal population with mean 1 and variance 2, and let X21, X22, p , X2n2 be a random sample from a second normal pop1 ulation with mean 2 and variance 2. Assume that both normal populations are 2 2 independent. Let S 1 and S 2 be the sample variances. Then the ratio 2 F has an F distribution with n1 inator degrees of freedom.

Using and reusing a component can save lots of development time. In this section, a new Web project is created that uses the DataComponent assembly. For the purposes of these examples, a second copy of Visual Studio .NET is opened, and a new Web site is created.

1 denom-

This result is based on the fact that (n1 1)S 2/ 2 is a chi-square random variable with n1 1 1 1 degrees of freedom, that (n2 1)S 2 2 is a chi-square random variable with n2 1 degrees 2 2 of freedom, and that the two normal populations are independent. Clearly under the null 2 hypothesis H0: 2 S 2 S 2 has an Fn1 1,n2 1 distribution. This is the basis of 1 2 1 2 the ratio F0 the following test procedure.

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