Problem 6 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A data set \(x_{1}, x_{2}, \ldots, x_{n}\) is given, modeled as realization of a sample \(X_{1}, X_{2}, \ldots, X_{n}\) from an \(N(\mu, 1)\) distribution. Suppose there are sample statistics \(L_{n}=g\left(X_{1}, \ldots, X_{n}\right)\) and \(U_{n}=h\left(X_{1}, \ldots, X_{n}\right)\) such that $$ \mathrm{P}\left(L_{n}<\mu

Step-by-Step Solution

Answer

a) Yes, \( \tilde{L}_n = 1 \), \( \tilde{U}_n = 22 \). c) \( \theta = 1-\mu \) has the interval \((-4, 3)\). d) A CI for \( \theta = \mu^2 \) isn't feasible directly.

Step 1: Understand the Definition of the Confidence Interval

A confidence interval provides a range that, with a certain probability (called confidence level), contains the true parameter. Here, \( P(L_n < \mu < U_n) = 0.95 \) indicates that this is a 95% confidence interval for \( \mu \). For any linear transformation of \( \mu \), this interval can be shifted or scaled accordingly.

Step 2: Part a - Transform the Confidence Interval

Given \( \theta = 3\mu + 7 \), for \( L_n < \mu < U_n \), we perform the transformation: \[ \tilde{L}_n = 3L_n + 7 \] \[ \tilde{U}_n = 3U_n + 7 \] Now substitute into the probability: \[ P(\tilde{L}_n < \theta < \tilde{U}_n) = P(3L_n + 7 < 3\mu + 7 < 3U_n + 7) = P(L_n < \mu < U_n) = 0.95 \] This shows that the transformed values \( \tilde{L}_n \) and \( \tilde{U}_n \) form a 95% confidence interval for \( \theta \).

Step 3: Part b - Confidence Interval for θ in terms of ln and un

Given \( \theta = 3\mu + 7 \) and \((l_n, u_n) = (−2,5)\): \[ \tilde{L}_n = 3(-2) + 7 = -6 + 7 = 1 \] \[ \tilde{U}_n = 3(5) + 7 = 15 + 7 = 22 \] Thus, the 95% confidence interval for \( \theta \) is \((1, 22)\).

Step 4: Part c - Change of Variable and Interval Construction

Given \( \theta = 1 - \mu \), redefine the bounds: \[ \tilde{L}_n = 1 - U_n \] \[ \tilde{U}_n = 1 - L_n \] Using \( (l_n, u_n) = (−2,5) \) we calculate: \[ \tilde{L}_n = 1 - 5 = -4 \] \[ \tilde{U}_n = 1 - (-2) = 3 \] Hence, for \( \theta = 1 - \mu \), the 95% confidence interval is \((-4, 3)\).

Step 5: Part d - Squared Transformation

Given \( \theta = \mu^2 \), the confidence bounds \( L_n \) and \( U_n \) don't transform linearly due to squaring. Squaring is non-linear and non-monotonic, making a direct confidence interval tricky to establish. Without further information (e.g., distribution of \( \mu^2 \)), a straightforward transformation isn't feasible for non-linear cases.

Key Concepts

Normal Distribution

In statistics, the normal distribution is a fundamental concept, often referred to as the Gaussian distribution. It is a symmetric, bell-shaped curve that describes how data is distributed around a mean. The characteristics of a normal distribution include:

• Symmetry: The left and right sides of the distribution are mirror images.
• Mean, median, and mode all coincide at the center of the distribution.
• Defined by two parameters: mean (\(\mu\)) and standard deviation (\(\sigma\)), with the standard deviation indicating the spread of data.

When a data set is said to be normally distributed, it means that most of the data points cluster around the mean (\(\mu\)) with fewer data points further away. This assumption allows for various statistical methods and calculations to be applied, such as confidence intervals. A significant feature is that about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Confidence Level

A confidence level in statistics indicates the probability that a confidence interval captures the true population parameter. It is expressed as a percentage, most commonly 95% or 99%, reflecting how confident we are in our interval estimates.

• A 95% confidence level means that if we were to take 100 different samples and compute intervals, approximately 95 of them would contain the true parameter value.
• This does not imply that there is a 95% probability that the parameter is within a single interval you've calculated, as the true value is fixed and does not vary like the sample statistics.

Therefore, a confidence level in our context informs us about the reliability of the interval. It essentially reflects the frequency with which the true parameter is expected to lie within an interval framed around the sample statistics when repeated sampling is hypothetically possible.

Statistical Transformations

Statistical transformations refer to operations applied to data or statistics to alter its scale or distribution to facilitate analysis or interpretation. In the context of confidence intervals, transformations are often applied to rescale or shift the interval appropriately.

• Linear transformations, such as multiplication or addition, adjust the interval endpoints proportionally. For example, if \(\theta = 3\mu + 7\), then the interval for \(\theta\) is transformed by applying the same linear operations to the interval bounds.
• Non-linear transformations, such as squaring or taking logarithms, require cautious handling as they can alter the distribution shape and center, complicating the direct transformation of confidence intervals.

Understanding and applying the correct transformation ensures that intervals maintain their properties, like the confidence level, post-transformation, enabling accurate data interpretation.

Sample Statistics

Sample statistics are numerical values that summarize or describe features of a sample data set. They serve as estimators for population parameters, facilitating inferences about the larger population.

• Examples include the sample mean (\(\bar{X}\)), sample variance (\(s^2\)), and sample standard deviation (\(s\)).
• Sample statistics are derived from a subset of the population, using methods like random sampling to ensure representation and minimize bias.

In practice, these statistics are critical when constructing confidence intervals. For instance, the sample mean might be used as the point estimate for the population mean.
The accuracy of these statistics in approximating population parameters is improved as sample size increases, due to the Law of Large Numbers, which states that sample statistics converge to population parameters as the sample size grows. This convergence underlies their use in creating reliable confidence intervals.