91Ó°ÊÓ

The article "Objective Measurement of the Stretchability of Mozzarella Cheese" \((J\). Texture Stud., 1992: 185-194) reported on an experiment to investigate how the behavior of mozzarella cheese varied with temperature. Consider the accompanying data on \(x=\) temperature and \(y=\) elongation (\%) at failure of the cheese. [Note: The researchers were Italian and used real mozzarella cheese, not the poor cousin widely available in the United States.] $$ \begin{array}{r|rrrrrrr} x & 59 & 63 & 68 & 72 & 74 & 78 & 83 \\ \hline y & 118 & 182 & 247 & 208 & 197 & 135 & 132 \end{array} $$ a. Construct a scatter plot in which the axes intersect at \((0,0)\). Mark \(0,20,40,60,80\), and 100 on the horizontal axis and \(0,50,100,150,200\), and 250 on the vertical axis. b. Construct a scatter plot in which the axes intersect at \((55,100)\), as was done in the cited article. Does this plot seem preferable to the one in part (a)? Explain your reasoning. c. What do the plots of parts (a) and (b) suggest about the nature of the relationship between the two variables?

Step by step solution

01

Prepare Data for Plotting

List the given values of temperature (\(x\)) and elongation (\(y\)) in order to visualize the relationship. Make sure to note the provided data as pairs for plotting: \((59, 118),\ (63, 182),\ (68, 247),\ (72, 208),\ (74, 197),\ (78, 135),\ (83, 132)\).

02

Plot Scatter Plot for Part (a)

Draw a scatter plot with the axes intersecting at \((0,0)\). Label the horizontal axis with temperature values: \(0, 20, 40, 60, 80\), and the vertical axis with elongation values: \(0, 50, 100, 150, 200, 250\). Plot each point according to the data pairs.

03

Plot Scatter Plot for Part (b)

Draw a second scatter plot with the axes intersecting at \((55, 100)\). This means shifting the origin so the x-axis starts at 55 and the y-axis starts at 100. Plot each data point accordingly and ensure the axes are marked correctly relative to this new intersection.

04

Compare the Two Scatter Plots

Compare the two scatter plots to evaluate which better represents the relationship between temperature and elongation. Consider how clearly each plot exhibits trends or patterns in the data, and note how the shifted axis might make the trend more apparent.

05

Analyze the Relationship

Based on the scatter plots, note any observable trends or patterns. The plots suggest a non-linear relationship where elongation increases with temperature up to a certain point (around \(x=68\)), then decreases. This indicates that elongation has an optimal temperature after which performance declines.

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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 fundamental tool in statistical data visualization. It allows us to see the relationships between two quantitative variables by representing them as individual points on a Cartesian coordinate system. For this exercise, each point represents a pair consisting of temperature (\( x \)) and elongation (\( y \)) values of mozzarella cheese.
To construct a scatter plot, one must first identify the two sets of data that will be represented on the x-axis and y-axis. Here, temperature is plotted on the x-axis, while elongation percentage is plotted on the y-axis. The position of each data point is determined by a pair of numerical values from these two datasets.

• The scatter plot in part (a) starts with the origin at \((0, 0)\), providing a straightforward visualization that includes any possible zero values in the data. A point like (59, 118), for example, shows a temperature of 59 degrees and an elongation of 118%.
• The scatter plot in part (b) shifts the origin to \((55, 100)\), which can sometimes offer improved clarity by centering around the specific range of interest—especially when data does not span near-zero values.

This method is particularly useful for identifying patterns or correlations, as it visually expresses how one variable behaves relative to changes in another.

Temperature and Elongation Relationship

Understanding the relationship between temperature and elongation is crucial for identifying the material properties. In the context of mozzarella cheese, elongation refers to the cheese's capacity to stretch without breaking. Exploring how this property changes with temperature provides insights into its behavior under different conditions.
From the data provided, as the temperature varies, we notice different elongation responses. Initially, increasing temperature enhances the cheese's elongation, showing better stretchability. However, after reaching a certain temperature peak, further increases result in reduced elongation. This could imply a thermal limit to mozzarella's optimal stretchability.

• At about 68 degrees, the cheese achieves maximum elongation, emphasizing a possible ideal thermal condition for stretchability.
• Beyond this temperature, a decline in elongation suggests the cheese may lose its structural integrity and become less cohesive.

These observations help us understand how temperature influences physical properties, guiding practical applications like cooking or further industrial processing.

Graph Interpretation

Graph interpretation involves analyzing visual data representations to derive meaningful information. It requires one to observe not just the plotted points, but also the tendencies they exhibit.
When looking at the scatter plots for mozzarella cheese, several insights emerge. For part (a), the data appearing at \((0,0)\) emphasizes absolute measurements, but may crowd the lower end of the scale if the data does not extend near zero. Meanwhile, part (b), starting from \((55, 100)\), provides a focused view around the most significant variation zone. It highlights changes in elongation more clearly, given the data's limited temperature range.

• Identify peaks and troughs: Recognize where the maximum and minimum values occur.
• Understand patterns: Notice the general trends, like increasing and then decreasing elongation as temperature changes.

A carefully selected scale can make the data's story more apparent, revealing trends that might be less obvious otherwise.

Non-linear Relationship

The term 'non-linear relationship' describes a connection between two variables that do not follow a straight line when plotted. Instead of a constant rate of change, the relationship may curve, indicating complexity in the data.
For the mozzarella cheese example, the relationship between temperature and elongation is non-linear. As temperature rises, elongation initially increases, suggesting an enhancing effect. After a maximum point around 68 degrees, elongation diminishes, revealing a downturn in stretchability at higher temperatures. Such behavior suggests:

• A peak or optimal point can occur, marking the best condition for a variable's performance.
• The need to consider multiple factors that could influence behavior, beyond a simple one-to-one correspondence.

Understanding such relationships can have practical implications, like determining ideal conditions for processes, or improving product qualities based on environmental changes.