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A scientist collected data using four different instruments. She organized her results in a table, with the measurements from each instrument in a separate column. The actual value is 5.5. Which column shows both high precision and high accuracy?

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$1$[/tex] & [tex]$2$[/tex] & [tex]$3$[/tex] & [tex]$4$[/tex] \\
\hline
5.4 & 4.9 & 8.9 & 3.2 \\
\hline
5.7 & 6.2 & 4.1 & 2.9 \\
\hline
5.6 & 7.3 & 4.3 & 3.1 \\
\hline
5.7 & 5.1 & 9.2 & 3.1 \\
\hline
5.6 & 3.7 & 3.3 & 2.8 \\
\hline
\end{tabular}

A. 2
B. 3
C. 1
D. 4



Answer :

GinnyAnswer
To determine which column indicates both high precision and high accuracy, we need to compute two important metrics for each instrument's measurements:

1. Accuracy: This can be assessed using the Mean Absolute Error (MAE), which measures the average magnitude of the errors between the observed values and the actual value.
2. Precision: This is measured using the Standard Deviation, which indicates how much the values are spread out from their mean.

Given the actual value is 5.5, we use the following data:
- Instrument 1: [5.4, 5.7, 5.6, 5.7, 5.6]
- Instrument 2: [4.9, 6.2, 7.3, 5.1, 3.7]
- Instrument 3: [8.9, 4.1, 4.3, 9.2, 3.3]
- Instrument 4: [3.2, 2.9, 3.1, 3.1, 2.8]

### Calculating Metrics:

For Instrument 1:
- Mean Absolute Error (MAE): [tex]\( 0.13999999999999985 \)[/tex]
- Standard Deviation (SD): [tex]\( 0.10954451150103316 \)[/tex]

For Instrument 2:
- Mean Absolute Error (MAE): [tex]\( 1.06 \)[/tex]
- Standard Deviation (SD): [tex]\( 1.22245654319489 \)[/tex]

For Instrument 3:
- Mean Absolute Error (MAE): [tex]\( 2.38 \)[/tex]
- Standard Deviation (SD): [tex]\( 2.546841180757057 \)[/tex]

For Instrument 4:
- Mean Absolute Error (MAE): [tex]\( 2.48 \)[/tex]
- Standard Deviation (SD): [tex]\( 0.14696938456699082 \)[/tex]

### Comparing Metrics:

To determine which instrument has the highest accuracy, we look for the smallest MAE. Among the values provided:
- Instrument 1 has the smallest MAE [tex]\( 0.13999999999999985 \)[/tex].

To determine which instrument has the highest precision, we look for the smallest standard deviation. Among the values provided:
- Instrument 1 also has the smallest standard deviation [tex]\( 0.10954451150103316 \)[/tex].

### Conclusion:
Instrument 1 (Column 1) exhibits both the highest accuracy (smallest MAE) and the highest precision (smallest SD). Therefore, the answer is:
C. 1

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