This exercise is intended to check the statement: "If we...

This exercise is intended to check the statement: "If we were to compute confidence intervals a large number of times, we would find that approximately $(1-\alpha) \times 100 \%$ of the time, the computed intervals would contain the true value of the parameter." (a) Assuming that the mean is unknown and that the variance is known, find the $90 \%$ confidence interval for the mean of a Gaussian random variable with $n=10$. (b) Generate 500 batches of 10 zero-mean, unit-variance Gaussian random variables. and determine the associated confidence intervals. Find the proportion of confidence intervals that include the true mean (which by design is zero). Is this in agreement with the confidence level $1-\alpha=.90$ ? (c) Repeat part b using exponential random variables with mean one. Should the proportion of intervals including the true mean be given by $1-\alpha$ ? Explain.

This exercise is intended to check the statement: "If we were to compute confidence intervals a large number of times, we would find that approximately $(1-\alpha) \times 100 \%$ of the time, the computed intervals would contain the true value of the parameter."
(a) Assuming that the mean is unknown and that the variance is known, find the $90 \%$ confidence interval for the mean of a Gaussian random variable with $n=10$.
(b) Generate 500 batches of 10 zero-mean, unit-variance Gaussian random variables. and determine the associated confidence intervals. Find the proportion of confidence intervals that include the true mean (which by design is zero). Is this in agreement with the confidence level $1-\alpha=.90$ ?
(c) Repeat part b using exponential random variables with mean one. Should the proportion of intervals including the true mean be given by $1-\alpha$ ? Explain.

Key Concepts

Robustness and Distributional Assumptions

When confidence intervals derived under the assumption of normality are applied to data from a different distribution (such as exponential), the actual coverage probability might deviate from the intended level. This concept highlights the importance of distributional assumptions in statistical inference; if the data do not conform to the assumed model, the performance and reliability of the confidence intervals may be compromised. Understanding this robustness is crucial when applying parametric methods to real-world data that may exhibit non-normal behavior.

Simulation Studies in Statistical Inference

Simulation studies involve generating repeated random samples from a known probability distribution to assess the performance of statistical methods, such as confidence interval calculations. By simulating multiple batches of data and computing the corresponding intervals, statisticians can empirically evaluate the coverage probability—the proportion of intervals that successfully capture the true parameter—thereby confirming the theoretical confidence level.

Confidence Intervals

A confidence interval is a range calculated from sample data that is likely to contain the true value of an unknown population parameter. It is associated with a confidence level (such as 90% or 95%), which reflects the long-run frequency at which the computed intervals would capture the true parameter if the same procedure were repeated many times. The frequentist interpretation does not imply a probability for the parameter itself but rather describes the performance of the interval estimation procedure over repeated experiments.

Gaussian Distribution with Known Variance

The Gaussian (or normal) distribution is a fundamental probability distribution in statistics characterized by its bell-shaped curve. When the variance is known, and the mean is unknown, the sample mean's distribution is also normal, making it straightforward to derive confidence intervals using the standard normal distribution. This setting simplifies analysis since the standardization of the sample mean produces a statistic that exactly follows a known distribution, allowing for direct calculation of critical values.

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