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Magazine Chapter 7: Problem 19
Suppose that the random variable \(X\) has a log normal distribution with parameters \(\theta=1.5\) and \(\omega=0.8\). A sample of size \(n=15\) is drawn from this distribution. Find the standard error of the sample median of this distribution with the bootstrap method using \(n_{n}=200\) bootstrap samples.
Short Answer
Expert verified
The standard error is the standard deviation of the bootstrap medians.
Step by step solution
01
Understand the Log-Normal Distribution
The random variable \(X\) is log-normally distributed, which means that \( \ln(X) \sim \mathcal{N}(\theta, \omega^2) \). So, it's important to understand that when \(X\) is distributed this way, the natural logarithm of \(X\), \(\ln(X)\), follows a normal distribution with mean \(\theta = 1.5\) and variance \(\omega^2 = 0.64\).
02
Draw the Original Sample
Draw a sample of size \(n=15\) from the log-normal distribution. Use the parameters \(\theta=1.5\) and \(\omega=0.8\) to generate the original sample data \(X_1, X_2, \ldots, X_{15}\). This is the basis for all the bootstrap samples.
03
Generate Bootstrap Samples
Take a bootstrap sample by randomly drawing \(15\) observations with replacement from the original sample. This process should be repeated \(n_{n} = 200\) times to create 200 bootstrap samples, \(X_1^*, X_2^*, \ldots, X_{15}^*\).
04
Calculate the Median for Each Bootstrap Sample
For each of the 200 bootstrap samples, calculate the sample median. Let the median of the \(i^{th}\) bootstrap sample be called \(\text{median}_i^*\) for \(i = 1, 2, \ldots, 200\).
05
Determine the Bootstrap Distribution of the Medians
Compile all 200 medians obtained from the bootstrap samples to form the bootstrap distribution of medians. Analyze how the median varies among the bootstrap samples.
06
Compute the Standard Error of the Sample Median
Calculate the standard deviation of the bootstrap distribution of sample medians. This standard deviation is the bootstrap estimate of the standard error of the sample median.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Log-Normal Distribution
A log-normal distribution describes a situation where a variable grows multiplicatively, meaning it's affected by random percent changes. The log-normal distribution is unique because if you take the natural logarithm of a log-normal variable, it converts into a normal distribution. In simpler terms, if you have a log-normal variable, like our random variable \(X\), then \(\ln(X)\) is approximately normal with specific parameters.
- For our exercise, we have parameters \(\theta = 1.5\) and \(\omega = 0.8\), meaning that if \(\ln(X)\) is normally distributed with mean \(\theta\) and variance \(\omega^2 = 0.64\).
- This is powerful because many natural phenomena behave in ways that are better modeled by log-normal distributions, particularly when values can鈥檛 be negative and they vary over orders of magnitude.
Random Sampling
Random sampling is fundamental for gathering a representative subset from a larger population. It ensures each member of the population has an equal chance of being included. This methodology is crucial for making sound inferences.
In our exercise, we drew a sample of size \(n = 15\) from a log-normal distribution. Randomness in sampling guarantees that the sample reflects the population without bias, an essential aspect when applying further statistical methods like the bootstrap.
In our exercise, we drew a sample of size \(n = 15\) from a log-normal distribution. Randomness in sampling guarantees that the sample reflects the population without bias, an essential aspect when applying further statistical methods like the bootstrap.
- The process involves selecting samples in such a way that every possible sample configuration has an equal probability.
- This guarantees the generalizability of the conclusions drawn from the sample back to the entire population.
Standard Error
The standard error is crucial when evaluating the reliability and precision of a sample statistic. It represents the standard deviation of the sampling distribution of a statistic, often used with parameters like the mean or median.
In our context, we employ the standard error to comprehend the variability of the sample median obtained from repeated sampling.
In our context, we employ the standard error to comprehend the variability of the sample median obtained from repeated sampling.
- The concept is particularly significant when applying the bootstrap method, which estimates the standard error through resampling, providing a practical approach even when traditional assumptions are hard to check.
- Our problem required computing the standard error by assessing the standard deviation of 200 bootstrap sample medians. This technique effectively conveys the expected variability in the median.
Sample Median
The sample median is the middle value of a dataset when it is ordered from smallest to largest, providing a measure of central tendency. The median is less affected by skewed data or outliers than the mean, making it a robust statistic.
Given its effectiveness, the sample median is a pivotal statistic in describing the central position of a sample. Each bootstrap sample of our exercise involved calculating this median to construct a distribution from which variability could be analyzed.
Given its effectiveness, the sample median is a pivotal statistic in describing the central position of a sample. Each bootstrap sample of our exercise involved calculating this median to construct a distribution from which variability could be analyzed.
- With 200 bootstrap samples, the median of each reflects how the sample median might behave if we sampled the population many times.
- By investigating this bootstrap distribution of medians, we obtain a clearer picture of the expected variability and central tendency of the sample.
