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Sagot :
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
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
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