Due To The Variation Of Environmental Conditions And Imperfe

Due To The Variation Of Environmental Conditions And Imperfections In

Due to the variation of environmental conditions and imperfections in the measurement method, a single measurement will not produce the exact distance d. The scientist takes n measurements of the distance and uses the sample average to estimate the true distance. From past records of these measurements, the scientist knows the variance of a single measurement is 4 m². How many measurements should the scientist make so that the chance that his estimate differs from the true distance by more than 0.5 m is at most 0.05?

Paper For Above instruction

Accurate measurement of physical quantities such as distance is fundamental in scientific research and engineering applications. However, environmental variations and measurement imperfections introduce variability that affects precision. When repeated measurements are taken, the average is used to estimate the actual value, assuming that such averaging reduces the effect of randomness. In this context, determining the necessary number of measurements to achieve a specific confidence level is essential. Specifically, the problem involves estimating the number of measurements required such that the probability that the estimated mean differs from the true distance by more than a given margin is below a certain threshold.

Given that the variance of a single measurement is 4 m², and the goal is to ensure that the probability the estimate differs by more than 0.5 meters is at most 0.05, we analyze this problem through the lens of statistical inference, particularly leveraging properties of the sampling distribution of the mean. The key principle is the Central Limit Theorem (CLT), which states that with a sufficiently large sample size, the distribution of the sample mean approximates a normal distribution regardless of the underlying distribution, provided the original measurements are independent and identically distributed.

Mathematically, let X₁, X₂, ..., Xₙ denote the individual measurements, each with variance σ² = 4 m². The sample mean, \(\bar{X}\), has an expected value \(E[\bar{X}] = \mu\) and variance \(\text{Var}(\bar{X}) = \sigma^2 / n\). Consequently, \(\bar{X}\) is normally distributed approximately as \(N(\mu, \sigma^2/n)\) for large n.

To find the minimum number of measurements n, we specify the confidence requirement: the probability that \(|\bar{X} - \mu| > 0.5\) must be at most 0.05. Using the properties of the normal distribution, this translates to the statement that the two-tailed probability outside of ±0.5 must be less than or equal to 0.05.

The critical value \(z_{0.975}\) for a standard normal distribution corresponding to a cumulative probability of 0.975 (because 0.025 in each tail) is approximately 1.96. Therefore, the following inequality must hold:

\[ P(|\bar{X} - \mu| > 0.5) = 2 \times P(\bar{X} - \mu > 0.5) \leq 0.05 \]

Or equivalently,

\[ P\left( |\bar{X} - \mu| > 0.5 \right) = 2 \times P \left( Z > \frac{0.5}{\sigma / \sqrt{n}} \right) \leq 0.05 \]

where Z is a standard normal variable. The inequality simplifies to:

\[ \frac{0.5}{\sigma / \sqrt{n}} \geq 1.96 \]

Substituting \(\sigma^2 = 4\) gives \(\sigma = 2\). Rearranging the inequality:

\[ 0.5 / (2 / \sqrt{n}) \geq 1.96 \]

\[ 0.5 \times \sqrt{n} / 2 \geq 1.96 \]

\[ 0.25 \times \sqrt{n} \geq 1.96 \]

\[ \sqrt{n} \geq \frac{1.96}{0.25} \]

\[ \sqrt{n} \geq 7.84 \]

Squaring both sides:

\[ n \geq 7.84^2 \]

\[ n \geq 61.47 \]

Therefore, the scientist should take at least 62 measurements to ensure, with at least 95% confidence, that the estimate of the true distance will not differ from the true value by more than 0.5 meters.

Conclusion

This analysis demonstrates that, considering the known measurement variance, a sample size of at least 62 measurements is necessary. This ensures the accuracy of the average estimate within the specified margin with high confidence. Applying such statistical principles in experimental design enhances measurement reliability and confidence, fundamental aspects in scientific investigation.

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