A Statistician Wants To Estimate The Mean Height In Meters

A Statistician Wants To Estimate The Mean Height H In Meters Of A Po

A statistician wants to estimate the mean height h (in meters) of a population, based on n independent samples \(X_1, X_2, \ldots, X_n\), chosen uniformly from the entire population. He uses the sample mean \( \bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n} \) as the estimate of h. We also assume that he knows \( \text{Var}(X_i) = 0.5 \) (meter\(^2\)). How large should n be so that the variance of \( \bar{X}_n \) is at most \( 10^{-4} \)? Please use the weak law of large numbers, i.e., determine n so that the estimate is within 0.05 meters from h, with probability at least 0.9.

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The goal of this analysis is to determine an appropriate sample size \( n \) that ensures the sample mean height \( \bar{X}_n \) closely approximates the true mean height \( h \) of the population. The problem involves understanding variance constraints and probability bounds based on the weak law of large numbers (WLLN). By addressing these components, we can establish the minimum \( n \) required to meet the specified accuracy and confidence level.

The initial step involves utilizing the properties of the sample mean, particularly its variance. Given that each \( X_i \) is independently drawn from the population with a known variance \( \text{Var}(X_i) = 0.5 \), the variance of the sample mean \( \bar{X}_n \) can be directly calculated as:

\[

\text{Var}(\bar{X}_n) = \frac{\text{Var}(X_i)}{n} = \frac{0.5}{n}

\]

This relation indicates that increasing the sample size reduces the variance of the estimate, thus improving precision.

To ensure the variance of \( \bar{X}_n \) does not exceed \( 10^{-4} \), the following inequality must hold:

\[

\frac{0.5}{n} \leq 10^{-4}

\]

Solving for \( n \):

\[

n \geq \frac{0.5}{10^{-4}} = \frac{0.5}{0.0001} = 5000

\]

Thus, sampling at least 5000 individuals guarantees that the variance of the sample mean does not surpass the specified threshold, providing a first assurance regarding the estimate's accuracy.

Next, to utilize the weak law of large numbers for probabilistic guarantees, specifically to assert that \( \bar{X}_n \) is within 0.05 meters of \( h \) with at least 90% probability, we consider the variance and apply Chebyshev's inequality. Chebyshev's inequality states that for any random variable \( Y \) with finite variance:

\[

P(|Y - \mathbb{E}[Y]| \geq a) \leq \frac{\text{Var}(Y)}{a^2}

\]

Applying this to \( \bar{X}_n \):

\[

P(|\bar{X}_n - h| \geq 0.05) \leq \frac{\text{Var}(\bar{X}_n)}{(0.05)^2}

\]

Substituting the known variance:

\[

\leq \frac{0.5 / n}{0.0025} = \frac{0.5}{0.0025 n}

\]

To ensure the probability that \( |\bar{X}_n - h| \) exceeds 0.05 meters is at most 0.1 (complement of at least 0.9 probability), set:

\[

\frac{0.5}{0.0025 n} \leq 0.1

\]

Solving for \( n \):

\[

n \geq \frac{0.5}{0.0025 \times 0.1} = \frac{0.5}{0.00025} = 2000

\]

Therefore, sampling at least 2000 individuals guarantees, via Chebyshev's inequality, that the estimate \( \bar{X}_n \) lies within 0.05 meters of \( h \) with at least 90% probability.

In summary, to meet both the variance constraint (\( \text{Var}(\bar{X}_n) \leq 10^{-4} \)) and the probabilistic accuracy requirement (\( P(|\bar{X}_n - h|

This rigorous approach illustrates how understanding variance and inequalities such as Chebyshev's, alongside the laws of large numbers, guides the determination of adequate sample sizes in statistical estimation problems. Proper sample sizing ensures reliable and precise estimations with high confidence, which is fundamental in fields such as biostatistics, social sciences, and environmental studies.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Loève, M. (1977). Probability Theory (4th ed.). Springer.
  • Ross, S. M. (2014). Introduction to Probability and Statistics (11th ed.). Academic Press.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses (3rd ed.). Springer.
  • Hogg, R. V., McKean, J., & Craig, A. T. (2013). Introduction to Mathematical Statistics (7th ed.). Pearson.
  • Frye, D. (2010). Applied Statistics. McGraw-Hill.
  • Kendall, M. G., & Stuart, A. (1973). The Advanced Theory of Statistics (4th ed.). Griffin.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman.

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