2. Describing Data with Tables and Graphs
Visualizing Qualitative vs. Quantitative Data
3:26 minutesProblem 10.5.8
Textbook Question
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.
Sound Intensity The table lists intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity.
Verified step by step guidance1
Step 1: Begin by plotting a scatterplot of the given data. Use the 'Sound Intensity' values as the x-axis and the 'Scale Value' as the y-axis. This will help visualize the relationship between the two variables.
Step 2: Analyze the scatterplot to determine the general trend or pattern in the data. Look for whether the data points suggest a linear, quadratic, logarithmic, exponential, or power relationship.
Step 3: Fit different mathematical models to the data. For each model (linear, quadratic, logarithmic, exponential, and power), calculate the corresponding equation using regression techniques. For example, for a linear model, use the formula y = mx + b, where m is the slope and b is the y-intercept.
Step 4: Evaluate the goodness-of-fit for each model using statistical measures such as the coefficient of determination (R²). The model with the highest R² value is typically the best fit for the data.
Step 5: Once the best-fitting model is identified, ensure that the model is appropriate for the scope of the given data. Interpret the model in the context of the problem, and confirm that it aligns with the observed trend in the scatterplot.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Scatterplot
A scatterplot is a graphical representation of two variables, where each point represents an observation in the dataset. It helps visualize the relationship between the variables, allowing for the identification of patterns, trends, or correlations. In this context, plotting sound intensity against scale value will help determine how these two variables interact.
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Mathematical Models
Mathematical models are equations or functions that describe the relationship between variables. In this case, the focus is on linear, quadratic, logarithmic, exponential, and power models. Each model has distinct characteristics and is suitable for different types of data patterns, making it essential to choose the one that best fits the observed data in the scatterplot.
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Sound Intensity and Decibel Scale
Sound intensity refers to the power per unit area carried by sound waves, often measured in watts per square meter. The decibel scale is a logarithmic scale used to quantify sound intensity levels, where each increase of 10 dB represents a tenfold increase in intensity. Understanding this relationship is crucial for interpreting the data in the context of sound measurement and its representation in the scatterplot.
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