Finding the Best Model
In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.
Detecting Fraud Leading digits of check amounts are often analyzed for the purpose of detecting fraud. The accompanying table lists frequencies of leading digits from checks written by the author (an honest guy).

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Step 1: Construct a scatterplot by plotting the leading digits (1 through 9) on the x-axis and their corresponding frequencies on the y-axis. This visual representation will help identify the pattern or trend in the data.

Step 2: Observe the shape of the scatterplot to determine which type of mathematical model might fit best. For example, if the points decrease rapidly and then level off, a logarithmic or power model might be appropriate; if the points form a curve opening downward or upward, a quadratic model might fit; if the points decrease steadily, a linear or exponential model might be suitable.

Step 3: Consider the context of the data. Since the data represents frequencies of leading digits, it is known from Benford's Law that the distribution often follows a logarithmic pattern, where lower digits occur more frequently than higher digits.

Step 4: To confirm the best model, calculate or fit each candidate model (linear, quadratic, logarithmic, exponential, power) to the data using regression techniques and compare their goodness-of-fit measures, such as R-squared values.

Step 5: Select the model with the best fit (highest R-squared and reasonable residuals) for the given data, keeping in mind the model should only be used within the scope of the provided data.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Scatterplot Construction

A scatterplot is a graphical representation of data points plotted on a coordinate plane, showing the relationship between two variables. It helps visualize patterns, trends, or correlations, which is essential for selecting an appropriate mathematical model.

Model Selection and Types

Choosing the best-fitting model involves comparing different mathematical functions—linear, quadratic, logarithmic, exponential, and power—to see which best describes the data trend. Each model type has distinct characteristics and fits different data patterns.

Frequency Distribution and Fraud Detection

Frequency distribution shows how often each leading digit appears in the data. Analyzing these frequencies can reveal patterns consistent with natural data (like Benford's Law) or anomalies that may indicate fraud, making it crucial to understand the expected distribution.