91Ó°ÊÓ

The article" Objective Measurement of the Stretchability of Mozzarella Cheese" (J. of Texture Studies, 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. $$ \begin{array}{r|rrrrrrr} x & 59 & 63 & 68 & 72 & 74 & 78 & 83 \\ \hline y & 118 & 182 & 247 & 208 & 197 & 135 & 132 \end{array} $$ a. Construct a scatterplot 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 scatterplot 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 the data for plotting

Identify and list the points to be plotted for both scatterplots. You have temperature values \(x = [59, 63, 68, 72, 74, 78, 83]\) and elongation values \(y = [118, 182, 247, 208, 197, 135, 132]\). These will be your coordinate pairs, like \((59, 118), (63, 182)\), and so on.

02

Construct scatterplot with axes at (0,0)

Start with a graph where the x-axis begins at 0 and increments by 20 up to 100. The y-axis should begin at 0 and increment by 50 up to 250. Plot the given data points in this coordinate system. On this graph, the points will spread out according to their actual values on both axes.

03

Construct scatterplot with axes at (55,100)

Adjust the graph so the x-axis starts at 55 and the y-axis starts at 100. The x-axis increments remain similar, but you start from 55 instead of 0. Similarly, the y-axis increments start from 100 instead of 0. Re-plot the same data points on this new coordinate system. Observe how the shift in axes changes the appearance of the data spread.

04

Compare the two scatterplots

Examine how the data is displayed differently in each plot. The first plot (axes at (0,0)) shows the overall increase directly from zero, whereas the second plot (axes at (55,100)) better centers the data, making it easier to see patterns or trends in the data right where the data range begins and ends.

05

Interpret the nature of the relationship

With both graphs, assess the relationship between temperature and elongation. Look for patterns, such as a rise in elongation with temperature, followed by a peak and then a decrease. Determine if this suggests a linear, quadratic, or some other type of relation between the variables. The second plot may make it more apparent if it centers the data better.

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Key Concepts

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

Data Visualization

Imagine you have data that you need to clearly present. One effective way to do this is through data visualization. Data visualization uses visual elements like scatterplots to make understanding complex data easier. For example, in our original exercise, you have temperature values and elongation percentages of mozzarella cheese.
These values can be paired to form coordinate points for a scatterplot.

• Each point represents how much the cheese stretches at a certain temperature.
• By plotting these points, you can visually assess their relationship.
• Data visualization is crucial in identifying trends, patterns, and potential outliers within the data.

By initially plotting the data with axes intersecting at (0,0), you get a straightforward view of all values. This approach helps in displaying the full range of temperatures and elongation measurements without any shifts. It's the simplest form, showing the base, or origin, of both x and y axes. However, it might not emphasize specific trends due to the lack of focus on the actual data range.

Statistical Analysis

Once your data is visualized, it's time to dive deeper with statistical analysis. Using scatterplots as a tool, you can begin interpreting the data's meaning. By adjusting the scatterplot, such as placing the axes at (55,100), you can focus more closely on where your data actually lies.
This adjustment is not just for aesthetics; it can enhance interpretation.

• The data points will spread more evenly across the graph, allowing for clearer recognition of patterns.
• This shift may highlight the trend of increasing elongation up to a certain temperature, then eventually decreasing.
• Statistical analysis involves looking for these trends, signifying potential underlying patterns or relationships.

These patterns can suggest whether the relationship between temperature and cheese elongation is linear, indicating consistent change, or non-linear, indicating varied changes.

Temperature and Elongation Relationship

The relationship between temperature and elongation can be understood better through visualized data and statistical insights. The scatterplots reveal important characteristics of cheese behavior under varying temperatures.
At certain temperatures, the mozzarella shows increased elongation; however, elongation decreases as temperature continues to rise past a peak.

• This indicates a possible non-linear, like quadratic, relationship between the temperature and elongation.
• Such relationships can suggest an optimal temperature range for maximum stretching.
• Understanding these patterns is essential for applications, such as in food science, where knowing the physical properties of cheese regarding temperature can influence industrial processing methods.

Therefore, by focusing on the patterns revealed in the scatterplots, one grasps the complexity of temperature and elongation behavior, enhancing the understanding of the material's properties.