Question

Let \(X_{1}, \ldots, X_{n}\) be a random sample from a \(N(0,1)\) distribution. Then the probability that the random interval \(\bar{X} \pm t_{\alpha / 2, n-1}(s / \sqrt{n})\) traps \(\mu=0\) is \((1-\alpha)\). To verify this empirically, in this exercise, we simulate \(m\) such intervals and calculate the proportion that trap 0, which should be "close" to \((1-\alpha)\). (a) Set \(n=10\) and \(m=50\). Run the \(\mathrm{R}\) code mat=matrix (rnorm \((\mathrm{m} * \mathrm{n}), \mathrm{n} \overline{\mathrm{col}=\mathrm{n}})\) which generates \(m\) samples of size \(n\) from the \(N(0,1)\) distribution. Each row of the matrix mat contains a sample. For this matrix of samples, the function below computes the \((1-\alpha) 100 \%\) confidence intervals, returning them in a \(m \times 2\) matrix. Run this function on your generated matrix mat. What is the proportion of successful confidence intervals? (b) Run the following code which plots the intervals. Label the successful intervals. Comment on the variability of the lengths of the confidence intervals.

Short answer

The short answer will be specific numbers that could not be determined without the actual results of running the code. However, the general form of the short answer will be a statement like: 'The proportion of successful confidence intervals was approximately \', along with comments on the observed variability of confidence interval lengths.

Step-by-step solution

Set Parameters and Generate Matrix

Set the parameters \(n = 10\) and \(m = 50\). Then run the provided R code to generate an \(m \times n\) matrix, 'mat', of random values from an \(N(0,1)\) distribution. Each row of this matrix represents one of the \(m\) samples of size \(n\).

Calculate Confidence Intervals

Next, run the provided function on 'mat'. This function calculates the \((1-\alpha) 100\%\) confidence intervals for each of the \(m\) samples, and returns a new \(m \times 2\) matrix. Each row of this matrix represents the lower and upper bounds of the confidence interval for the corresponding sample from 'mat'.

Evaluate Proportion of Successful Confidence Intervals

To find the proportion of successful confidence intervals, count how many of the confidence intervals contain the value 0 (because the population mean \(\mu\) is 0 for the \(N(0,1)\) distribution). The proportion of successful intervals is the number of successful intervals divided by the total number of intervals (\(m\)).

Run Plotting Code

Run the provided code to plot each of the \((1-\alpha) 100\%\) confidence intervals. Successful intervals (those containing 0) should be labelled.

Comment on Variability of Confidence Interval Lengths

Observe the plotted confidence intervals. Their lengths can be computed as upper bound - lower bound. Comment on the variability of these lengths, noting whether there is a lot of consistency in interval length, or whether it varies considerably from one sample to the next.

Key concepts

Understanding the N(0,1) Distribution

The notation N(0,1) refers to a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. This specific distribution is also known as the standard normal distribution and serves as a foundation in statistics.

A key feature of the N(0,1) distribution is its symmetry about the mean, meaning that half of the values will be less than zero and the other half greater. The distribution is bell-shaped and the probability of observing a value within one standard deviation from the mean is approximately 68.2%, within two standard deviations is about 95.4%, and within three standard deviations is about 99.7%. These percentages are often used to describe the amount of data falling within certain intervals and are crucial when discussing confidence intervals.

When working with any normal distribution, it's critical to understand the properties associated with it, like the empirical rule and the z-scores, which indicate how many standard deviations away from the mean a point in the distribution lies. For the standard normal distribution, the z-score is equal to the value of the observation itself, due to mean being zero and standard deviation being one.

Calculating Confidence Intervals

Confidence intervals are essential tools in statistics used to estimate the reliability of an estimate. For instance, a 95% confidence interval suggests that if we were to take 100 different samples and compute the interval for each of them, we would expect about 95 of those intervals to contain the true population parameter.

To calculate a confidence interval, you need a point estimate (like the sample mean), and you consider how it might vary around the population mean (μ). The formula for a confidence interval often contains the term t_{α/2, n-1}, which is the value from the t-distribution that reflects the desired confidence level (1-α) and the sample size (n). The expression (s / √n) is the standard error of the sample, showing the estimate's precision. Multiply these together to get the 'margin of error', and add/subtract it from the sample mean to find the interval bounds.

• The wider the interval, the more uncertainty it conveys about the estimate.
• Narrow intervals suggest higher precision but require larger sample sizes or lower variability to achieve.
• Choosing the right confidence level, usually 95% or 99%, balances reliability with precision.

The reliability of a confidence interval is verified statistically by the percentage of intervals that include the true population parameter, as illustrated in the textbook exercise.

R Programming for Statistics

R is a powerful language and environment for statistical computing and graphics. It plays a significant role in data analysis, partly because it's open-source and has a large and active community contributing packages for various statistical tasks.

The R programming environment is particularly adept for statistics because it includes numerous built-in functions for standard procedures. For example, in the context of the exercise, rnorm is a fundamental function that generates random variates with a given distribution—in this case, the N(0,1) distribution. We work with matrices in R to handle large datasets efficiently, and a function can be written to apply statistical methods across rows or columns of the matrix, as was done to calculate confidence intervals for multiple samples.
The simulation approach demonstrated in the exercise allows students to understand statistical concepts empirically. By running the simulation multiple times, one can see how often the computed confidence intervals capture the population parameter, thereby gaining an intuitive grasp of the confidence level in practice. This hands-on experience is an effective way to learn and validate statistical concepts.