Combining two service-center rates
Two fictional service centers report activity for the same month. Center A completed 80 out of 100 cases successfully. Center B completed 30 out of 50 cases successfully. Cases are distinct and use the same definition of success. A report writer averages eighty percent and sixty percent to claim a combined seventy-percent success rate. Another writer wants to compare the observed rate with a target of seventy-five percent. No claim is made that the sample is random or that either center caused an improvement. The question concerns only arithmetic summaries of the supplied records. A separate planning scenario asks what would happen if ten additional unsuccessful cases were corrected to successful cases without changing the total number of cases. Keep that hypothetical change separate from the original observations.
1. Calculate the observed combined success rate.
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There are 110 successes in 150 cases, giving 110/150=73⅓%. The unweighted seventy-percent mean gives equal weight to centers with different case totals.
2. Compare with the seventy-five-percent target.
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The shortfall is 75-73⅓=1⅔percentage points. This is a difference between rates, not a relative percent-growth calculation.
3. Calculate the rate in the stated correction scenario.
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Successes become 120 while the denominator remains 150, so the hypothetical rate is 80%. The original observed rate remains 73⅓%.