Answer :
Let's discuss and solve the problem step-by-step to determine which team provided the most accurate measurement and which provided the most precise measurement.
### Definitions
1. Accuracy: Accuracy denotes how close a measured value is to the actual (true) value. In this context, it will refer to how close the teams' measured periods are to the reliable measured period of 0.410 s.
2. Precision: Precision represents the consistency or repeatability of measurements. It refers to how close the measured values are to each other, indicating the extent of variability.
### Given Data
- Reliable measurement: [tex]\(0.410 \, \text{s}\)[/tex]
- Team [tex]\(A\)[/tex]: [tex]\(0.51 \, \text{s} \pm 0.05 \, \text{s}\)[/tex]
- Team [tex]\(B\)[/tex]: Between [tex]\(0.380 \, \text{s}\)[/tex] and [tex]\(0.390 \, \text{s}\)[/tex]
- Team [tex]\(C\)[/tex]: [tex]\(0.310 \, \text{s} \pm 8.0\% \)[/tex]
- Team [tex]\(D\)[/tex]: [tex]\(0.360 \, \text{s}\)[/tex]
### Calculations for Accuracy
Team A:
- Measurement: [tex]\(0.51 \, \text{s}\)[/tex]
- Error bounds: [tex]\(0.51 \, \text{s}\)[/tex]
Team B:
- Measurements: Between [tex]\(0.380 \, \text{s}\)[/tex] and [tex]\(0.390 \, \text{s}\)[/tex]
Team C:
- Measurement: [tex]\(0.310 \, \text{s}\)[/tex]
- Error percentage: [tex]\(8.0\%\)[/tex]
- Error value: [tex]\(0.310 \times 0.08 = 0.0248 \, \text{s}\)[/tex]
Team D:
- Measurement: [tex]\(0.360 \, \text{s}\)[/tex]
Now, let's compare the measurements with the reliable measurement to determine accuracy (absolute differences with the true value):
- Team A: [tex]\( |0.51 - 0.410| = 0.10 \)[/tex]
- Team B:
- Lower bound: [tex]\( |0.380 - 0.410| = 0.03 \)[/tex]
- Upper bound: [tex]\( |0.390 - 0.410| = 0.02 \)[/tex]
- Closest difference: [tex]\(0.02\)[/tex] (choosing the minimum absolute error)
- Team C: [tex]\( |0.310 - 0.410| = 0.10 \)[/tex]
- Team D: [tex]\( |0.360 - 0.410| = 0.05 \)[/tex]
### Most Accurate Measurement
The smallest absolute errors are:
- Team B: [tex]\(0.02\)[/tex]
So, Team B provided the most accurate measurement.
### Calculations for Precision
Precision is based on the consistency of the measurements around the mean.
- Team A: Precision: [tex]\(0.05 \, \text{s}\)[/tex]
- Team B: Range divided by 2: [tex]\( (0.390 - 0.380)/2 = 0.005 \, \text{s}\)[/tex]
- Team C: Precision: [tex]\(0.0248 \, \text{s}\)[/tex]
- Team D: No precision error given: Assume [tex]\(0 \, \text{s}\)[/tex]
### Most Precise Measurement
The smallest precision errors are:
- Team D: [tex]\(0 \, \text{s}\)[/tex]
So, Team D provided the most precise measurement.
### Summary
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{team} & \begin{tabular}{c} \text{most accurate} \\ \text{measurement} \end{tabular} & \begin{tabular}{c} \text{most precise} \\ \text{measurement} \end{tabular} \\ \hline \text{A} & & \\ \hline \text{B} & \bigcirc & \\ \hline \text{C} & & \\ \hline \text{D} & & \bigcirc \\ \hline \end{tabular} \][/tex]
- The most accurate measurement is by Team B.
- The most precise measurement is by Team D.
### Definitions
1. Accuracy: Accuracy denotes how close a measured value is to the actual (true) value. In this context, it will refer to how close the teams' measured periods are to the reliable measured period of 0.410 s.
2. Precision: Precision represents the consistency or repeatability of measurements. It refers to how close the measured values are to each other, indicating the extent of variability.
### Given Data
- Reliable measurement: [tex]\(0.410 \, \text{s}\)[/tex]
- Team [tex]\(A\)[/tex]: [tex]\(0.51 \, \text{s} \pm 0.05 \, \text{s}\)[/tex]
- Team [tex]\(B\)[/tex]: Between [tex]\(0.380 \, \text{s}\)[/tex] and [tex]\(0.390 \, \text{s}\)[/tex]
- Team [tex]\(C\)[/tex]: [tex]\(0.310 \, \text{s} \pm 8.0\% \)[/tex]
- Team [tex]\(D\)[/tex]: [tex]\(0.360 \, \text{s}\)[/tex]
### Calculations for Accuracy
Team A:
- Measurement: [tex]\(0.51 \, \text{s}\)[/tex]
- Error bounds: [tex]\(0.51 \, \text{s}\)[/tex]
Team B:
- Measurements: Between [tex]\(0.380 \, \text{s}\)[/tex] and [tex]\(0.390 \, \text{s}\)[/tex]
Team C:
- Measurement: [tex]\(0.310 \, \text{s}\)[/tex]
- Error percentage: [tex]\(8.0\%\)[/tex]
- Error value: [tex]\(0.310 \times 0.08 = 0.0248 \, \text{s}\)[/tex]
Team D:
- Measurement: [tex]\(0.360 \, \text{s}\)[/tex]
Now, let's compare the measurements with the reliable measurement to determine accuracy (absolute differences with the true value):
- Team A: [tex]\( |0.51 - 0.410| = 0.10 \)[/tex]
- Team B:
- Lower bound: [tex]\( |0.380 - 0.410| = 0.03 \)[/tex]
- Upper bound: [tex]\( |0.390 - 0.410| = 0.02 \)[/tex]
- Closest difference: [tex]\(0.02\)[/tex] (choosing the minimum absolute error)
- Team C: [tex]\( |0.310 - 0.410| = 0.10 \)[/tex]
- Team D: [tex]\( |0.360 - 0.410| = 0.05 \)[/tex]
### Most Accurate Measurement
The smallest absolute errors are:
- Team B: [tex]\(0.02\)[/tex]
So, Team B provided the most accurate measurement.
### Calculations for Precision
Precision is based on the consistency of the measurements around the mean.
- Team A: Precision: [tex]\(0.05 \, \text{s}\)[/tex]
- Team B: Range divided by 2: [tex]\( (0.390 - 0.380)/2 = 0.005 \, \text{s}\)[/tex]
- Team C: Precision: [tex]\(0.0248 \, \text{s}\)[/tex]
- Team D: No precision error given: Assume [tex]\(0 \, \text{s}\)[/tex]
### Most Precise Measurement
The smallest precision errors are:
- Team D: [tex]\(0 \, \text{s}\)[/tex]
So, Team D provided the most precise measurement.
### Summary
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{team} & \begin{tabular}{c} \text{most accurate} \\ \text{measurement} \end{tabular} & \begin{tabular}{c} \text{most precise} \\ \text{measurement} \end{tabular} \\ \hline \text{A} & & \\ \hline \text{B} & \bigcirc & \\ \hline \text{C} & & \\ \hline \text{D} & & \bigcirc \\ \hline \end{tabular} \][/tex]
- The most accurate measurement is by Team B.
- The most precise measurement is by Team D.
