Question

In an attempt to determine any possible relationship between the year in which a U.S. penny was minted and its current mass (in grams), students weighed an assortment of pennies and obtained the following data. $$\begin{array}{lllllll}\hline 1968 & 1973 & 1977 & 1980 & 1982 & 1983 & 1985 \\\\\hline 3.11 & 3.14 & 3.13 & 3.12 & 3.12 & 2.51 & 2.54 \\\3.08 & 3.06 & 3.10 & 3.11 & 2.53 & 2.49 & 2.53 \\\3.09 & 3.07 & 3.06 & 3.08 & 2.54 & 2.47 & 2.53 \\\\\hline\end{array}$$ What valid conclusion(s) might they have drawn about the relationship between the masses of the pennies within a given year and from year to year?

Short answer

After calculating and comparing the average masses, it becomes apparent that the mass of pennies decreased significantly after the year 1982. This observation provides a strong indication that there is a relationship between the U.S. pennies' current mass and their minting year. However, to precisely determine the nature of this relationship, a more detailed statistical analysis would be necessary.

Step-by-step solution

Identify Data Points

Observe the data provided regarding the mass of pennies from different years. Each row in the matrix corresponds to a set of data points from a series of years ranging from 1968 to 1985. The accompanying mass values for these years are given in grams.

Calculate the Average Mass

A sensible way to approach this data is to calculate the average mass of pennies for each year, which would provide a more concrete point for comparison. To do this, use the formula for the arithmetic mean, which is the sum of all values divided by the number of values. Calculate this for each year.

Compare Average Mass from Year to Year

After determining the average mass for pennies from each year, compare these averages. This will give an idea if there's an overall trend or particular pattern in the mass of pennies over the years.

Draw Conclusions

Based on the comparison of average masses, draw valid conclusions about the relationship between the year a penny was minted and its current mass. Also, observe any changes in the mass of pennies within a given year and from year to year.

Key concepts

Arithmetic Mean

The arithmetic mean is a mathematical concept used to summarize a set of data by providing the 'average' value. This average is obtained by adding up all the numbers in a data set and dividing by the total number of observations.

In the context of analyzing penny masses from different years, determining the arithmetic mean for each year offers several advantages:

• It simplifies complex data by distilling it into a single number, making it easier to compare across years.
• It helps to identify trends or patterns in data over time, such as whether pennies became lighter or heavier.
• It reduces the effect of anomalies, such as a single penny that might be unusually heavy or light, by averaging out the values.

To calculate the arithmetic mean for a given year's penny mass, sum all the mass values for that year and divide by the number of mass measurements taken. This simple calculation provides a clearer picture of the mass characteristics for pennies minted in that year.

Mass Measurement

Mass measurement is a fundamental process in chemistry and data analysis, particularly when exploring physical properties of materials over time. Accurate mass measurements are essential for understanding changes in materials, like the evolution of minting techniques reflected in the mass of pennies.

When measuring mass, precision is crucial. It ensures that the data collected is reliable and can be used to draw meaningful comparisons. This precision is often achieved through the use of calibrated balances capable of measuring minute differences in mass. Key points in maintaining accuracy in mass measurement include:

• Using a well-calibrated balance to ensure measurements are precise.
• Taking multiple measurements to account for variability and using the arithmetic mean for greater reliability.
• Handling the samples consistently to avoid introducing errors from contaminants or handling practices.

Through these practices, students can confidently explore and analyze the patterns in data, such as the mass of pennies from different mint years.

Scientific Conclusion Drawing

Scientific conclusion drawing is the process of interpreting data to make reasoned judgments or hypotheses about observed patterns. This skill is crucial in connecting data analysis to wider contexts or theories, such as analyzing changes in minting processes over time.

In the example of analyzing penny masses, conclusions can be drawn about the historical changes in manufacturing techniques or materials used. To draw scientific conclusions effectively, one should:

• Clearly define the comparison basis, like identifying specific years with significant mass differences.
• Utilize calculated averages to support or refute hypotheses about patterns or trends.
• Consider external factors that may influence the data, such as the introduction of new minting materials or technology shifts during specific years.

Drawing scientific conclusions involves not only analyzing data but also interpreting it in a way that considers historical, technological, and material changes over time. This holistic approach ensures conclusions are not only statistically valid but also scientifically meaningful.