Questions: Four research teams measured the density of a new alloy, and what each team wrote in its team notebook is shown in the table below. Suppose a later and more reliable measurement gives 8.10 g / cm^3 for the density of the same alloy. Decide which of the earlier measurements was the most accurate, and which was the most precise.

Four research teams measured the density of a new alloy, and what each team wrote in its team notebook is shown in the table below.
Suppose a later and more reliable measurement gives 8.10 g / cm^3 for the density of the same alloy. Decide which of the earlier measurements was the most accurate, and which was the most precise.

Transcript text: Four research teams measured the density of a new alloy, and what each team wrote in its team notebook i, own in the table below. Suppose a later and more reliable measurement gives $8.10 \mathrm{~g} / \mathrm{cm}^{3}$ for the density of the same alloy. Decide which of the earlier measurements was the most accurate, and which was the most precise.

Solution

Solution Steps

Step 1: Understanding Accuracy and Precision

Accuracy refers to how close a measurement is to the true or accepted value. Precision refers to how close multiple measurements are to each other, regardless of whether they are close to the true value.

Step 2: Analyzing Team A's Measurement

Team A reported a density of $7.6 \, \text{g/cm}^3 \pm 7.0\%$. The range of this measurement is: \[ 7.6 \, \text{g/cm}^3 \pm 0.532 \, \text{g/cm}^3 \Rightarrow [7.068, 8.132] \, \text{g/cm}^3 \] The midpoint is $7.6 \, \text{g/cm}^3$, which is $0.5 \, \text{g/cm}^3$ away from the true value of $8.10 \, \text{g/cm}^3$.

Step 3: Analyzing Team B's Measurement

Team B reported a range between $8.15 \, \text{g/cm}^3$ and $8.25 \, \text{g/cm}^3$. The midpoint is: \[ \frac{8.15 + 8.25}{2} = 8.20 \, \text{g/cm}^3 \] The midpoint is $0.10 \, \text{g/cm}^3$ away from the true value of $8.10 \, \text{g/cm}^3$.

Step 4: Analyzing Team C's Measurement

Team C reported a density of $9.10 \, \text{g/cm}^3$, which is $1.00 \, \text{g/cm}^3$ away from the true value of $8.10 \, \text{g/cm}^3$.

Step 5: Analyzing Team D's Measurement

Team D reported a density of $8.60 \, \text{g/cm}^3 \pm 0.25 \, \text{g/cm}^3$. The range of this measurement is: \[ [8.35, 8.85] \, \text{g/cm}^3 \] The midpoint is $8.60 \, \text{g/cm}^3$, which is $0.50 \, \text{g/cm}^3$ away from the true value of $8.10 \, \text{g/cm}^3$.

Step 6: Determining the Most Accurate Measurement

The most accurate measurement is the one closest to the true value of $8.10 \, \text{g/cm}^3$. Team B's midpoint of $8.20 \, \text{g/cm}^3$ is the closest, with a deviation of $0.10 \, \text{g/cm}^3$.

Step 7: Determining the Most Precise Measurement

The most precise measurement is the one with the smallest range. Team D's range is $0.50 \, \text{g/cm}^3$ ($8.60 \pm 0.25$), which is smaller than Team A's range of $1.064 \, \text{g/cm}^3$ and Team B's range of $0.10 \, \text{g/cm}^3$.