Several students measured the mass of a rabbit and recorded their measurements in a table.

\begin{tabular}{|l|l|}
\hline Student 1 & [tex]$1.6 \text{ kg} = 1,600 \text{ g}$[/tex] \\
\hline Student 2 & [tex]$2,300 \text{ g}$[/tex] \\
\hline Student 3 & Between [tex]$1,000$[/tex] and [tex]$1,500 \text{ g}$[/tex] \\
\hline Student 4 & [tex]$1,750 \text{ g}$[/tex] \\
\hline
\end{tabular}

The actual mass of the rabbit is [tex]$1,864.3 \text{ g}$[/tex]. Which statement best describes the accuracy and precision of their data?

A. Neither accurate nor precise
B. Accurate but not precise
C. Precise but not accurate
D. Both accurate and precise



Answer :

Let's go through the solution step-by-step to determine the accuracy and precision of the students' measurements.

First, let's compile the measurements:
- Student 1: [tex]\(1600\)[/tex] g
- Student 2: [tex]\(2300\)[/tex] g
- Student 3: [tex]\( \frac{1000 + 1500}{2} = 1250\)[/tex] g (since the measurement is between 1000 and 1500 g)
- Student 4: [tex]\(1750\)[/tex] g

So, the measurements are: [tex]\(1600, 2300, 1250, 1750\)[/tex] (in grams).

Next, we need to calculate the mean (average) of these measurements:
[tex]\[ \text{Mean Measurement} = \frac{1600 + 2300 + 1250 + 1750}{4} = \frac{6900}{4} = 1725 \text{ g} \][/tex]

Now, let's consider the accuracy, which is the difference between the mean measurement and the actual mass of the rabbit:
[tex]\[ \text{Actual Mass} = 1864.3 \text{ g} \][/tex]
[tex]\[ \text{Accuracy} = | \text{Mean Measurement} - \text{Actual Mass} | = |1725 - 1864.3| = 139.3 \text{ g} \][/tex]

Next, we need to evaluate the precision, which is the standard deviation of the measurements. For a set of measurements [tex]\(x_1, x_2, x_3, x_4\)[/tex], the standard deviation [tex]\( \sigma \)[/tex] is given by:
[tex]\[ \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + (x_3 - \mu)^2 + (x_4 - \mu)^2}{4 - 1}} \][/tex]
where [tex]\( \mu \)[/tex] is the mean measurement. Calculating each term:
[tex]\[ (x_1 - \mu)^2 = (1600 - 1725)^2 = (-125)^2 = 15625 \][/tex]
[tex]\[ (x_2 - \mu)^2 = (2300 - 1725)^2 = 575^2 = 330625 \][/tex]
[tex]\[ (x_3 - \mu)^2 = (1250 - 1725)^2 = (-475)^2 = 225625 \][/tex]
[tex]\[ (x_4 - \mu)^2 = (1750 - 1725)^2 = 25^2 = 625 \][/tex]

Summing these differences:
[tex]\[ 15625 + 330625 + 225625 + 625 = 572500 \][/tex]

Now, dividing by [tex]\(n-1 = 4-1 = 3\)[/tex]:
[tex]\[ \frac{572500}{3} \approx 190833.33 \][/tex]

Taking the square root gives the standard deviation:
[tex]\[ \sigma = \sqrt{190833.33} \approx 437.01 \text{ g} \][/tex]

Now, let's determine the classification based on accuracy and precision:
1. Accuracy: Given the accuracy threshold of 100g, our calculated accuracy of [tex]\(139.3\)[/tex] g indicates the measurements are not accurate.
2. Precision: Given the precision threshold of 200g, our calculated standard deviation of [tex]\(437.01\)[/tex] g indicates the measurements are not precise.

Given these conditions, the measurements are neither accurate nor precise.

Therefore, the best statement that describes the accuracy and precision of the data is:
A. Neither accurate nor precise