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The hypothesis-testing procedures that we have discussed in previous sections are designed for problems in which the population or probability distribution is known and the hypotheses involve the parameters of the distribution. Another kind of hypothesis is often encountered: we do not know the underlying distribution of the population, and we wish to test the hypothesis that a particular distribution will be satisfactory as a population model. For example, we might wish to test the hypothesis that the population is normal. We have previously discussed a very useful graphical technique for this problem called probability plotting and illustrated how it was applied in the case of a normal distribution. In this section, we describe a formal goodness-of- t test procedure based on the chi-square distribution.

Be sure to use the Delete method of the DataRow if changes are going to be sent back to the data store using the DataAdapter. If the Remove method of the DataTable.Rows collection is used, the DataRow will be completely removed from the DataTable, but there will be no deletion of the row at the data store.

The test procedure requires a random sample of size n from the population whose probability distribution is unknown. These n observations are arranged in a frequency histogram, having k bins or class intervals. Let Oi be the observed frequency in the ith class interval. From the hypothesized probability distribution, we compute the expected frequency in the ith class interval, denoted Ei. The test statistic is

(9-39)

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110 112 114 116 118 119 120 Create a Formula Apply Absolute and Relative Cell References Understanding Functions Apply a Function Total Cells with AutoSum Audit a Worksheet for Errors Create a Chart

In its simplest form, a DataRow can be edited by assigning new contents to DataRow. Using this method, however, triggers validation with each change. It is better to use the BeginEdit method of the DataRow, which will postpone validation until the EndEdit method is executed. When you use the BeginEdit method, the changes may be rolled back by executing the CancelEdit method. The following code is an example of editing a DataRow.

It can be shown that, if the population follows the hypothesized distribution, X 2 has, approx0 imately, a chi-square distribution with k p 1 degrees of freedom, where p represents the number of parameters of the hypothesized distribution estimated by sample statistics. This approximation improves as n increases. We would reject the hypothesis that the distribution of the population is the hypothesized distribution if the calculated value of the test statistic 2 2 0 ,k p 1. One point to be noted in the application of this test procedure concerns the magnitude 2 of the expected frequencies. If these expected frequencies are too small, the test statistic X0 will not re ect the departure of observed from expected, but only the small magnitude of the expected frequencies. There is no general agreement regarding the minimum value of expected frequencies, but values of 3, 4, and 5 are widely used as minimal. Some writers suggest that an expected frequency could be as small as 1 or 2, so long as most of them exceed 5. Should an expected frequency be too small, it can be combined with the expected frequency in an adjacent class interval. The corresponding observed frequencies would then also be combined, and k would be reduced by 1. Class intervals are not required to be of equal width. We now give two examples of the test procedure. EXAMPLE 9-12 A Poisson Distribution The number of defects in printed circuit boards is hypothesized to follow a Poisson distribution. A random sample of n 60 printed boards has been collected, and the following number of defects observed.

The mean of the assumed Poisson distribution in this example is unknown and must be estimated from the sample data. The estimate of the mean number of defects per board is the sample average, that is, (32 0 15 1 9 2 4 3) 60 0.75. From the Poisson

distribution with parameter 0.75, we may compute pi, the theoretical, hypothesized probability associated with the ith class interval. Since each class interval corresponds to a particular number of defects, we may nd the pi as follows: p1 p2 p3 p4 P1X P1X P1X P1X 02 12 22 32 e e e 1

Create objects Dim cnstr As String = integrated security=yes;database=northwind Dim cn As New SqlConnection(cnstr) Dim sql As String = Select CustomerID, CompanyName from customers Dim daCustomers As New SqlDataAdapter(sql, cn) Dim dt As New DataTable( Customers ) Execute daCustomers.Fill(dt) Assign the primary key. dt.PrimaryKey = New DataColumn() {dt.Columns( CustomerID )} Dim dr As DataRow = dt.Rows.Find( AAAAB ) dr.BeginEdit() dr( CustomerID ) = AAAAE

10.752 0 0! 10.752 1 1!

The expected frequencies are computed by multiplying the sample size n probabilities pi. That is, Ei npi. The expected frequencies follow:

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