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In statistics, the term 'Observed Frequency' refers to the actual count or number of occurrences of a particular event within a dataset or sample. For instance, when analyzing a group of people, the observed frequency would tell us how many individuals exhibit the characteristic or happen to fall under the category we're interested in. In the original exercise, the observed frequency was the number of college graduates in a sample who had student loans. Specifically, out of 80 college graduates, 48 had student loans. This figure, 48, is our observed frequency. It is what you can literally count or measure in your data. The clarity in recognizing the observed frequency helps ensure accurate analyses and conclusions. Being able to identify these frequencies is foundational for further computations, such as calculating proportions or comparing with expected outcomes.

The 'Observed Proportion' gives us a fraction or percentage that represents the portion of a total sample exhibiting a particular trait. It is a useful way to gauge the relative magnitude of an observed frequency against the entire group, making sizes more understandable and comparable.To determine the observed proportion, we use the formula:\[ \text{Observed Proportion} = \frac{\text{Observed Frequency}}{\text{Total Sample Size}} \]For the exercise provided, the observed proportion of college graduates with student loans is calculated by dividing the 48 graduates who took loans by the sample size of 80. This gives us:\[ \frac{48}{80} = 0.6 \text{ or } 60\% \]This percentage provides a straightforward representation of how common an event is within the sample. Observed proportions allow for easier comparisons between different datasets or samples, aiding in deeper analysis and insights.

The 'Expected Frequency' is a predicted count or estimate of the number of occurrences expected in a sample, assuming the measure like a probability is accurate. It is often calculated to compare against observed frequencies.To find the expected frequency, we multiply the total sample size by the estimated probability (proportion) of the occurrence. Using the formula:\[ \text{Expected Frequency} = \text{Probability} \times \text{Sample Size} \]In the context of our exercise, if the expected percentage of college graduates with loans is 64%, we calculate:\[ 0.64 \times 80 = 51.2 \]This means that, based on the estimated rate, about 51.2 graduates, in theory, should have student loans. It is an integral part of statistical analysis as it offers a baseline for understanding whether the observed occurrences deviate from what might be anticipated, helping identify potential anomalies or validate assumptions.