91Ó°ÊÓ

Penicillin was administered orally to five different horses, and the concentration of penicillin in the blood was determined after five different lengths of time. The following data on \(x=\) elapsed time (in hours) and \(y=\) penicillin concentration (in \(\mathrm{mg} / \mathrm{ml}\) ) appeared in the paper "Absorption and Distribution Patterns of Oral Phenoxymethyl Penicillin in the Horse" (Cornell Veterinarian [1983]: \(314-323\) ): 8 1 $$ \begin{array}{lrrrrr} \boldsymbol{x} & 1 & 2 & 3 & 6 \\ \boldsymbol{y} & 1.8 & 1.0 & 0.5 & 0.1 & 0.1 \end{array} $$ Construct scatterplots using the following variables. Which transformation, if any, would you recommend? a. \(x\) and \(y\) c. \(x\) and \(\sqrt{y}\) b. \(\sqrt{x}\) and \(y\) d. \(\sqrt{x}\) and \(\sqrt{y}\) e. \(x\) and \(\log (y)\) (values of \(\log (y)\) are \(0.26,0,-0.30\), \(-1\), and \(-1\) )

Step by step solution

01

Analysis of the given factors

The given factors are \(x\) = elapsed time in hours since penicillin was administered and \(y\) = penicillin concentration (\(\mathrm{mg} / \mathrm{ml}\)) after each elapsed time in the horse's bloodstream. These values are denoted in the table. We are asked to plot the scatterplots of \(x\) vs. \(y\), \(x\) vs. \(\sqrt{y}\), \(\sqrt{x}\) vs. \(y\), \(\sqrt{x}\) vs. \(\sqrt{y}\) and \(x\) vs. \(\log(y)\).

02

Construct scatterplots of \(x\) and \(y\)

Create a scatterplot where the horizontal axis represents time elapsed (\(x\)) and the vertical axis represents penicillin concentration (\(y\) \(\mathrm{mg} / \mathrm{ml}\)). Plot the given data point pairs. Observe the pattern of the scatterplot to draw conclusions about the relationship between \(x\) and \(y\).

03

Construct scatterplots of \(x\) and \(\sqrt{y}\)

This time, transform the \(y\) values by taking the square root and plot these transformed values against the original \(x\) values. Observe the pattern of the scatterplot to draw conclusions about the relationship between \(x\) and \(\sqrt{y}\).

04

Construct scatterplots of \(\sqrt{x}\) and \(y\)

Now transform the \(x\) values by taking the square root and plot these against the original \(y\) values. Observe the pattern of the scatterplot to draw conclusions about the relationship between \(\sqrt{x}\) and \(y\).

05

Construct scatterplots of \(\sqrt{x}\) and \(\sqrt{y}\)

Take the square root of both \(x\) and \(y\) values and then construct a scatterplot. Observe the pattern to analyze the relationship between \(\sqrt{x}\) and \(\sqrt{y}\).

06

Construct scatterplots of \(x\) and \(\log(y)\)

Transform \(y\) values by using a natural logarithm and plot these logarithmic values (\(\log(y)\)) against the \(x\) values. Observe the pattern of the scatterplot to draw conclusions about the relationship between \(x\) and \(\log(y)\).

07

Recommend a transformation

Based on the scatterplots constructed, recommend a transformation. This entails that, if the scatterplot indicates a linear relationship, no transformation is required. If the scatterplot forms a curve, either concave up or down, the data may be better represented by an exponential function, thus logarithmic transformation may be required. If the scatterplot forms a parabola, either facing upwards or downwards, the data may be better represented by a quadratic function, hence square or square root transformation might be required. Select the transformation that produces the clearest trend and minimizes the randomness.

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Key Concepts

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

Data Transformation

Data transformation is a crucial technique when analyzing data through scatterplots, especially when the relationship between the two variables isn't immediately clear. In this context, transforming the data involves altering the original values using mathematical operations, such as logarithms or square roots, to achieve a linear or more interpretable form. By transforming data:

• You can stabilize the variance across the data set.
• Enhance interpretability by making patterns more obvious.
• Make the data comply better with statistical models that assume linearity or normal distribution.

For example, using a logarithmic transformation on the penicillin concentration values can linearize an initially nonlinear trend. Similarly, applying a square root transformation might better align the relationship between time and concentration. The ultimate goal is to use the transformation that best fits the data trend and provides a straightforward interpretation with minimized error.

Linear Relationship

A linear relationship between two variables is one where a change in one variable is directly proportional to a change in another. In scatterplot analysis, identifying a linear relationship means the data points tend to align closely along a straight line. This suggests the response (penicillin concentration, in this case) changes at a constant rate as the time elapsed varies. If a scatterplot with transformed variables shows a linear pattern, it aids in simplification and ease of prediction. Sometimes, transformations, such as \(\log(y)\) or \(\sqrt{x}\), can reveal linear relationships that were not obvious in the raw data, helping researchers use linear models for analysis and prediction.

Quadratic Functions

Quadratic functions describe a parabolic relationship, where the response variable changes not at a constant rate, but rather accelerates or decelerates with respect to the predictor variable. Detecting such a pattern in a scatterplot indicates that neither a simple linear trend nor an exponential growth is the best fit. Instead, the scatterplot might suggest a curve that is increased or decreased by the square of the independent variable.If the relationship between time and penicillin concentration fits a parabolic shape, transformations involving squares or square roots may be optimal. For example, \(\sqrt{y}\) or \(\sqrt{x}\) transformations can showcase quadratic tendencies, making it easier to apply quadratic models and accurately describe the behavior of the variables.

Exponential Functions

Exponential functions depict scenarios where the rate of change in the response variable grows or decreases exponentially rather than linearly or quadratically. In this context, the penicillin concentration may drop rapidly and then level off as it might be absorbed or disseminated through the body over time. Scatterplots illustrating exponential trends may benefit from logarithmic transformations. For instance, transforming concentration values with \(\log(y)\) can often straighten a scatterplot with an exponential curve, making it appear linear and easier to analyze. The transformed scatterplot thus offers a new perspective, allowing more precise predictions utilizing linear regression or other statistical methods aligned to a newly-linearized trend.