To determine which statement best describes the data, we can follow these detailed steps:
1. Identify Objective:
- We need to assess whether the data are precise, accurate, and reproducible.
2. Explanation of Terms:
- Accurate: The data points lie close to the expected value, which is 130.
- Precise: The data points are close to each other.
- Reproducible: The data points should not all be identical; there should be some variation, but the data should still meet the criteria for accuracy or precision.
3. Given Data:
- Expected value: 130
- Observed values: 129, 131, 129, 132
4. Calculate the Mean:
- Mean of the observed data is calculated as follows:
[tex]\[
\text{Mean} = \frac{129 + 131 + 129 + 132}{4} = \frac{521}{4} = 130.25
\][/tex]
5. Check Accuracy:
- The difference between the expected value (130) and the calculated mean (130.25) is:
[tex]\[
|130 - 130.25| = 0.25
\][/tex]
- Since the difference is small (0.25), the data are considered accurate because they are close to the expected value of 130.
6. Calculate the Standard Deviation:
- Observed data: 129, 131, 129, 132
- Calculate the variance:
[tex]\[
\text{Variance} = \frac{(129 - 130.25)^2 + (131 - 130.25)^2 + (129 - 130.25)^2 + (132 - 130.25)^2}{4}
\][/tex]
- Simplify:
[tex]\[
= \frac{1.5625 + 0.5625 + 1.5625 + 3.0625}{4} = \frac{6.75}{4} = 1.6875
\][/tex]
- Standard Deviation (SD) is the square root of the variance:
[tex]\[
\text{SD} = \sqrt{1.6875} \approx 1.3
\][/tex]
7. Check Precision and Reproducibility:
- Precision: Because the standard deviation (1.3) is relatively low, the data points are considered precise.
- Reproducibility: Since the data points are not all identical and there is some variation, the data points are reproducible.
8. Conclusion:
- Given that the data points are accurate (mean is close to the expected value), precise (low standard deviation), and reproducible (variations exist but are within acceptable range), the best description is:
They are accurate and reproducible.
