11-1 Goodness-of-Fit 585 Rationale for the Test Statistic Examples 1 and 2 show that the x2 test statistic is a measure of the discrepancy between observed and expected frequencies. Simply summing the differences O - E between observed and expected values tells us nothing because that sum is always 0. Squaring the O - E values gives us a better statistic. (The reasons for squaring the O - E values are essentially the same as the reasons for squaring the x - x values in the formula for standard deviation.) The value of Σ1O - E22 measures only the magnitude of the differences, but we need to find the magnitude of the differences relative to what was expected. We need a type of average instead of a cumulative total. This relative magnitude is found through division by the expected frequencies, as in the test statistic Σ1O - E22>E. The theoretical distribution of Σ1O - E22>E is a discrete distribution because the number of possible values is finite. The distribution can be approximated by a chi-square distribution, which is continuous. This approximation is generally considered acceptable, provided that all expected values E are at least 5. (There are ways of circumventing the problem of an expected frequency that is less than 5, such as combining some categories so that all expected frequencies are at least 5. Also, there are different procedures that can be used when not all expected frequencies are at least 5.) The number of degrees of freedom reflects the fact that we can freely assign frequencies to k - 1 categories before the frequency for every category is determined. (Although we say that we can “freely” assign frequencies to k - 1 categories, we cannot have negative frequencies, nor can we have frequencies so large that their sum exceeds the total of the observed frequencies for all categories combined.) Goodness-of-Fit Test Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Statdisk 1. Click Analysis in the top menu. 2. Select Goodness-of-Fit from the dropdown menu. 3. Select Equal Expected Frequencies or Unequal Expected Frequencies. 4. Enter the desired significance level and select the column containing the observed frequencies. For Unequal Expected Frequencies also indicate if data are in the format of counts or proportions and select the column containing expected data. 5. Click Evaluate. StatCrunch 1. Click Stat in the top menu. 2. Select Goodness-of-fit from the dropdown menu, then select Chi-Square Test from the submenu. 3. Select the column with the observed frequencies. 4. Select the column containing the expected frequencies if expected frequencies are not all equal. Otherwise, click All cells in equal proportion. 5. Click Compute! Minitab 1. Click Stat in the top menu. 2. Select Tables from the dropdown menu and select Chi-Square Goodness-ofFit Test from the submenu. 3. Click Observed Counts and select the column containing the observed frequencies. 4. Under Test select Equal proportions if expected frequencies are all equal. For unequal expected frequencies or proportions, select Proportions specified by historical counts and select the column containing the expected frequencies or proportions. 5. Click OK. continued
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