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Chapter 5: Problem 60

Determining the age of an animal can sometimes be a difficult task. $$ \begin{array}{rrrrrrrrr} x & 0.25 & 0.25 & 0.50 & 0.50 & 0.50 & 0.75 & 0.75 & 1.00 \\ y & 700 & 675 & 525 & 500 & 400 & 350 & 300 & 300 \\ x & 1.00 & 1.00 & 1.00 & 1.00 & 1.25 & 1.25 & 1.50 & 1.50 \\ y & 250 & 230 & 150 & 100 & 200 & 100 & 100 & 125 \\ x & 2.00 & 2.00 & 2.50 & 2.75 & 3.00 & 4.00 & 4.00 & 5.00 \\ y & 60 & 140 & 60 & 50 & 10 & 10 & 10 & 10 \\ x & 5.00 & 5.00 & 5.00 & 6.00 & 6.00 & & & \\ y & 15 & 10 & 10 & 15 & 10 & & & \end{array} $$ Construct a scatterplot for this data set. Would you describe the relationship between age and canal length as linear? If not, suggest a transformation that might straighten the plot.

Short Answer

Expert verified
A scatterplot is constructed using the given data. By analyzing the pattern of points, the relationship between age and canal length can be determined. If the pattern doesn't appear to be a straight line, the relationship isn't linear. In this case, a logarithmic transformation of the age might be considered.

Step by step solution

01

Understanding the Data

The given data is a set of 'x' values representing the age of an animal and corresponding 'y' values representing the canal length. 'x' and 'y' are pairs of data to be plotted on a Cartesian plane.
02

Constructing the Scatterplot

Plot the points on a graph where the x-axis represents the age (x values) and the y-axis represents the canal length (y values). Each point on the graph represents a pair of x (age) and y (canal length). Use a free graphing software or a manual paper and pencil method to create the scatterplot.
03

Analyzing the Scatterplot

Observe the plotted points on the scatterplot. If the points create a pattern that appears to be a straight line, then the relationship between age and canal length can be described as linear. If not, we need to think about a possible transformation.
04

Suggesting a Transformation

If the points do not form a straight line pattern, then the relationship between age and canal length is not linear. There are several transformations we can consider, such as taking the logarithm of the x values (logarithmic transformation), the square root or the reciprocal. The choice depends on the pattern observed. In general, a logarithmic transformation can be effective if the scatterplot suggests a logarithmic pattern.

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Key Concepts

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

Data Analysis
Data analysis involves examining, cleaning, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. For effective data analysis, one must understand the nature and the source of the data, which in this case is the age of an animal (measured in years) and its corresponding canal length (in millimeters).

Before diving into the construction of scatterplots or any transformation, the first step in data analysis is to identify any patterns, correlations, or outliers within the data. This insight helps us to determine the best method for visualizing the information and if there are any further steps needed, such as data transformation, to make the data more interpretable. Measures such as the mean, median, range, and standard deviation can also be calculated to better understand the data's distribution.
Scatterplot Analysis
Scatterplot analysis is a visual method used to determine the possible relationship between two variables. By plotting each data pair as a point on a graph with an x and y-axis, patterns can emerge that indicate relationships such as linear, non-linear, or no correlation at all.

In the context of animal age and canal length, the scatterplot works as a tool to visually represent the data points and provide insights into the relationship between these two variables. When analyzing a scatterplot, look for the overall direction (positive, negative, or none), form (linear, curved, etc.), and strength (how closely the points fit a pattern) of the relationship. A linear relationship would show a scatterplot where points approximate a straight line. Variability in points would suggest a more complex relationship, one that might require transformation to properly analyze.
Data Transformation
Data transformation involves changing the data from its original form to a format that allows for better analysis and visualization. In the case of a scatterplot not showing a clear linear relationship, transformations can be applied to the x or y data to straighten the points into a line.

Common transformation techniques include logarithmic, square root, and reciprocal transformations. The choice of transformation is dependent on the pattern of the data. For instance, if the scatterplot suggests that the relationship between the variables is logarithmic鈥攎eaning that as one variable increases, there is a rapid increase in the other variable which then levels off鈥攁 logarithmic transformation can be applied to the x-values to linearize the data. This transformation helps to stabilize the variance across the data set, making trends more evident and analytical models more applicable.

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Most popular questions from this chapter

The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003): $$ \begin{array}{cc} & \begin{array}{l} \text { Number of } \\ \text { Transplants } \\ \text { Year } \end{array} & \text { (in thousands) } \\ \hline 1(1990) & 15.0 \\ 2 & 15.7 \\ 3 & 16.1 \\ 4 & 17.6 \\ 5 & 18.3 \\ 6 & 19.4 \\ 7 & 20.0 \\ 8 & 20.3 \\ 9 & 21.4 \\ 10 \text { (1999) } & 21.8 \\ \hline \end{array} $$ a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between \(y=\) number of transplants performed and \(x=\) year. Describe how the number of transplants performed has changed over time from 1990 to 1999 . b. Compute the 10 residuals, and construct a residual plot. Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.

The paper "Accelerated Telomere Shortening in Response to Life Stress" (Proceedings of the National Academy of Sciences [2004]: 17312-17315) described a study that examined whether stress accelerates aging at a cellular level. The accompanying data on a measure of perceived stress \((x)\) and telomere length \((y)\) were read from a scatterplot that appeared in the paper. Telomere length is a measure of cell longevity. $$ \begin{array}{rrrl} \begin{array}{l} \text { Perceived } \\ \text { Stress } \end{array} & \begin{array}{l} \text { Telomere } \\ \text { Length } \end{array} & \begin{array}{l} \text { Perceived } \\ \text { Stress } \end{array} & \begin{array}{l} \text { Telomere } \\ \text { Length } \end{array} \\ \hline 5 & 1.25 & 20 & 1.22 \\ 6 & 1.32 & 20 & 1.3 \\ 6 & 1.5 & 20 & 1.32 \\ 7 & 1.35 & 21 & 1.24 \\ 10 & 1.3 & 21 & 1.26 \\ 11 & 1 & 21 & 1.3 \\ 12 & 1.18 & 22 & 1.18 \\ 13 & 1.1 & 22 & 1.22 \\ 14 & 1.08 & 22 & 1.24 \\ 14 & 1.3 & 23 & 1.18 \\ 15 & 0.92 & 24 & 1.12 \\ 15 & 1.22 & 24 & 1.5 \\ 15 & 1.24 & 25 & 0.94 \\ 17 & 1.12 & 26 & 0.84 \\ 17 & 1.32 & 27 & 1.02 \\ 17 & 1.4 & 27 & 1.12 \\ 18 & 1.12 & 28 & 1.22 \\ 18 & 1.46 & 29 & 1.3 \\ 19 & 0.84 & 33 & 0.94 \\ \hline \end{array} $$ a. Compute the equation of the least-squares line. b. What is the value of \(r^{2}\) ? c. Does the linear relationship between perceived stress and telomere length account for a large or small proportion of the variability in telomere length? Justify your answer.

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

Cost-to-charge ratios (the percentage of the amount billed that represents the actual cost) for 11 Oregon hospitals of similar size were reported separately for inpatient and outpatient services. The data are $$ \begin{array}{lcc} \text { Hospital } & \text { Inpatient } & \text { Outpatient } \\ \hline \text { Blue Mountain } & 80 & 62 \\ \text { Curry General } & 76 & 66 \\ \text { Good Shepherd } & 75 & 63 \\ \text { Grande Ronde } & 62 & 51 \\ \text { Harney District } & 100 & 54 \\ \text { Lake District } & 100 & 75 \\ \text { Pioneer } & 88 & 65 \\ \text { St. Anthony } & 64 & 56 \\ \text { St. Elizabeth } & 50 & 45 \\ \text { Tillamook } & 54 & 48 \\ \text { Wallowa Memorial } & 83 & 71 \\ \hline \end{array} $$ a. Does there appear to be a strong linear relationship between the cost-to- charge ratio for inpatient and outpatient services? Justify your answer based on the value of the correlation coefficient and examination of a scatterplot of the data. b. Are any unusual features of the data evident in the scatterplot? c. Suppose that the observation for Harney District was removed from the data set. Would the correlation coefficient for the new data set be greater than or less than the one computed in Part (a)? Explain.

In the study of textiles and fabrics, the strength of a fabric is a very important consideration. Suppose that a significant number of swatches of a certain fabric are subjected to different "loads" or forces applied to the fabric. The data from such an experiment might look as follows: $$ \begin{aligned} &\text { Hypothetical Data on Fabric Strength }\\\ &\begin{array}{lccccccc} \hline \begin{array}{l} \text { Load } \\ \text { (lb/sq in.) } \end{array} & \mathbf{5} & \mathbf{1 5} & \mathbf{3 5} & \mathbf{5 0} & \mathbf{7 0} & \mathbf{8 0} & \mathbf{9 0} \\ \hline \begin{array}{l} \text { Proportion } \\ \text { failing } \end{array} & 0.02 & 0.04 & 0.20 & 0.23 & 0.32 & 0.34 & 0.43 \\ \hline \end{array} \end{aligned} $$ a. Make a scatterplot of the proportion failing versus the load on the fabric. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the loads and fit the line \(y^{\prime}=a+b\) (Load). What is the significance of a positive slope for this line? c. What proportion of the time would you estimate this fabric would fail if a load of \(60 \mathrm{lb} / \mathrm{sq}\) in. were applied? d. In order to avoid a "wardrobe malfunction," one would like to use fabric that has less than a \(5 \%\) chance of failing. Suppose that this fabric is our choice for a new shirt. To have less than a \(5 \%\) chance of failing, what would you estimate to be the maximum "safe" load in \(\mathrm{lb} / \mathrm{sq}\) in.?
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