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Chapter 1: Problem 59

Data points \((x, y)\) are given. (a) Draw a scatter plot of the data points. (b) Make semilog and log-log plots of the data. (c) Is a linear, power, or exponential function appropriate for modeling these data? (d) Find an appropriate model for the data and then graph the model together with a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {2} & {4} & {6} & {8} & {10} & {12} \\\ \hline y & {0.08} & {0.12} & {0.18} & {0.26} & {0.35} & {0.53} \\\ \hline\end{array}$$

Short Answer

Expert verified
Use plots to determine model; fit exponential or power function and graph it with data.

Step by step solution

01

Plot the Scatter Plot

To visualize your given data points, first create a scatter plot. Plot \(x\)-values on the horizontal axis and \(y\)-values on the vertical axis. Plot the points: \((2, 0.08), (4, 0.12), (6, 0.18), (8, 0.26), (10, 0.35), (12, 0.53)\). The scatter plot will help in understanding the initial distribution and the trend of your data.
02

Create a Semilog Plot

Convert the \(y\)-axis into a logarithmic scale to create a semilog plot. On graphing paper or graphing software that supports a semilog format, keep the \(x\)-axis linear and use log scale for \(y\)-axis. Plot the transformed points: \((2, \log_{10}(0.08)), (4, \log_{10}(0.12)), (6, \log_{10}(0.18)), (8, \log_{10}(0.26)), (10, \log_{10}(0.35)), (12, \log_{10}(0.53))\). This helps to determine if an exponential model fits the data.
03

Create a Log-Log Plot

Now, convert both \(x\)-axis and \(y\)-axis into logarithmic scales for a log-log plot. Use graph paper or software that allows log-log plotting. The transformed points are \((\log_{10}(2), \log_{10}(0.08)), (\log_{10}(4), \log_{10}(0.12)), (\log_{10}(6), \log_{10}(0.18)), (\log_{10}(8), \log_{10}(0.26)), (\log_{10}(10), \log_{10}(0.35)), (\log_{10}(12), \log_{10}(0.53))\). This plot can show if a power model is suitable.
04

Evaluate Model Appropriateness

Compare the linearity of the scatter points in the regular, semilog, and log-log plots. A straight line in the scatter plot indicates a linear model, the semilog plot suggests an exponential model if linear, and the log-log plot suggests a power model. Choose the model where your data align best in a straight line.
05

Determine the Best Fit Model

Based on the plots, if the data are linear in the semilog plot, an exponential model like \( y = ab^x \) should be fitted. If linear in the log-log plot, a power model like \( y = ax^b \) is appropriate. Choose the model that aligns closest to a straight line.
06

Fit and Graph the Model

For the selected model, calculate the best fit parameters using statistical fitting techniques like least squares. Graph the fitted model equation along with the original data scatter plot to visualize their alignment. For example, the model might look like \( y = a \, e^{bx} \) or \( y = ax^b \), depending on the choice.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot
A scatter plot is a basic and effective way to visualize data points on a two-dimensional graph. In a scatter plot, each data point is represented by a dot, with its position on the graph defined by its x-coordinate and y-coordinate. For example, consider the set of data points given:
  • (2, 0.08),
  • (4, 0.12),
  • (6, 0.18),
  • (8, 0.26),
  • (10, 0.35),
  • (12, 0.53)
To create a scatter plot from these data, plot the x-values on the horizontal axis and y-values on the vertical axis. This graphical representation helps to identify patterns, trends, and potential outliers in the data. A scatter plot can also suggest a type of model that might fit the data well, making it a foundational first step in regression analysis and data modeling.
Semilog Plot
A semilog plot is a type of graph used when one axis is set on a logarithmic scale, while the other remains linear. This format is particularly useful in exposing exponential relationships in data. To create a semilog plot, you take the logarithm of the y-values while keeping the x-values the same. For instance, use
  • logarithmic transformations like \(\log_{10}(0.08), \log_{10}(0.12),\)
  • \(\log_{10}(0.18)\),\(\log_{10}(0.26)\),\(\log_{10}(0.35)\), and \(\log_{10}(0.53)\)
This approach makes it straightforward to identify whether an exponential model might be appropriate, as data that align into a straight line on a semilog plot suggest such a model. This visualization simplifies the discernment of patterns in data that are not immediately apparent on a linear plot.
Log-Log Plot
A log-log plot transforms both axes of a graph into a logarithmic scale. This technique is beneficial when seeking to uncover power-law behaviors within data. By converting both the x and y coordinates with logarithm functions, the transformed points for our data set include:
  • (\(\log_{10}(2)\), \(\log_{10}(0.08)\)),
  • (\(\log_{10}(4)\), \(\log_{10}(0.12)\)),
  • (\(\log_{10}(6)\), \(\log_{10}(0.18)\)),
  • (\(\log_{10}(8)\), \(\log_{10}(0.26)\)),
  • (\(\log_{10}(10)\), \(\log_{10}(0.35)\)),
  • (\(\log_{10}(12)\), \(\log_{10}(0.53)\))
In this format, if the data points align into a straight line, it indicates a power function could be a suitable model to describe the relationship. This method is integral for situations where data exhibit multiplicative patterns.
Data Modeling
Data modeling involves finding mathematical models that capture the underlying patterns in a set of data. It encompasses choosing the type of function that best represents the data relationship, such as linear, exponential, or power models. In regression analysis:
  • A linear model is appropriate if data suggest a constant rate of change.
  • An exponential model fits best if the rate of change increases or decreases rapidly.
  • Power models are apt when data behave proportionally over different scales.
The approach starts by plotting the data in different forms (scatter, semilog, log-log) to visualize potential patterns and ends with fitting a mathematical equation that models these observed relationships, optimizing for best-fit parameters.
Best Fit Model
The best fit model is a mathematical equation that most accurately represents the pattern of data points. After evaluating various plot types, an analyst will choose a model where the data align closest to a straight line. For example, if data points are linear in a semilog plot, an exponential model might look like \( y = ab^x \). Conversely, if linear in a log-log plot, a power model such as \( y = ax^b \) would be considered. Statistical techniques, like least squares estimation, are used to fine-tune the model parameters. Finally, this model is plotted alongside the original data to visually compare its fit, assuring that the model effectively summarizes the data trends.

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Most popular questions from this chapter

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