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The sample range is a statistical measure that gives us an idea of the spread or dispersion in a data set. To calculate the sample range, you simply subtract the smallest value from the largest value in the data set. This range tells us how far apart the data points are from each other. In the context of the exercise we're looking at, which involves measurements of melting points of a biodegradable polymer, calculating the range helps to understand the variability in the melting temperatures across different samples.
The formula for the range is straightforward:

• Identify the smallest observation in the data set.
• Identify the largest observation in the data set.
• Subtract the smallest observation from the largest observation to get the range.

In our given data, the smallest observation was 180.3°C and the largest was 182.6°C. Therefore, the sample range was calculated as:\[ \text{Range} = 182.6 - 180.3 = 2.3 \text{°C} \]
This tells us that the melting point of the polymer samples varied by 2.3°C.

Sample standard deviation is a more refined measure of dispersion, building on the concept of variance. It tells us, on average, how far each data point is from the mean of the data set. Whereas variance gives us the squared differences, the standard deviation brings this back to the same units as the data by taking the square root.
The process to calculate it involves several steps:

• Calculate the mean (average) of the data set.
• Subtract the mean from each data point to find the deviations.
• Square each deviation.
• Find the average of these squared deviations (this is the variance).
• Take the square root of the variance to get the standard deviation.

In our example, the sample variance was about 0.471, so we took its square root to find the sample standard deviation:\[ s = \sqrt{0.471} \approx 0.686 \]
This standard deviation measures how much the melting point temperatures vary around the average. It's particularly useful in contexts like quality control, where maintaining consistent product properties is crucial. Understanding how sample standard deviation works improves our ability to analyze and interpret data effectively, highlighting significantly different observations.

Biodegradable polymers, like poly(3-hydroxybutyrate) (PHB), are materials that can break down naturally in the environment through the action of microorganisms. Unlike traditional plastics, they do not persist in the environment for long periods but instead decompose into natural substances like water, carbon dioxide, and biomass. This property makes them highly valuable from a sustainability perspective.
PHB, as mentioned in the problem, is derived from renewable resources and has characteristics such as biocompatibility and semi-crystallinity, which makes it suitable for various applications. However, despite these advantages, a significant downside is its cost of production. These materials are often more expensive to produce than conventional plastics, making widespread adoption challenging.
Some key characteristics of biodegradable polymers like PHB include:

• Environmental Friendliness: They reduce plastic waste and are better for the environment.
• Renewability: Often obtained from renewable resources, promoting sustainable usage.
• Decomposition: They break down into harmless substances, which minimize environmental impact.

Research and advancements in this field aim to overcome the cost barrier, enhancing both the production processes and material properties. Understanding biodegradable polymers is crucial as we move towards a more sustainable future, focusing on reducing environmental pollution and conserving non-renewable resources.