Estimating the Median Use the sample data listed in Exercise 1 “Bootstrap Requirements” to generate 1000 bootstrap samples, and find the median in each of those samples. After obtaining the 1000 sample medians, find the 95% confidence interval estimate of the population median by evaluating p2.5 and p97.5 from the sorted 1000 medians. Given that the sample times in Exercise 1 are from the 50 times in Data Set 20 “Alcohol and Tobacco in Movies” and those 50 times have a median of 5.5, how well did the bootstrap method work to create a “good” confidence interval?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to use the bootstrap method to estimate the 95% confidence interval for the population median based on the sample data provided. Bootstrap involves resampling the original data with replacement to create multiple samples and then calculating a statistic (in this case, the median) for each sample.
2. Step 2: Generate 1000 bootstrap samples. To do this, randomly select data points from the original sample (with replacement) to create a new sample of the same size as the original. Repeat this process 1000 times to create 1000 bootstrap samples.
3. Step 3: Calculate the median for each bootstrap sample. For each of the 1000 bootstrap samples, compute the median value. This will result in a distribution of 1000 medians.
4. Step 4: Sort the 1000 medians in ascending order. Once sorted, identify the 2.5th percentile (p2.5) and the 97.5th percentile (p97.5) of the sorted medians. These values represent the lower and upper bounds of the 95% confidence interval for the population median.
5. Step 5: Compare the confidence interval obtained using the bootstrap method to the known median of the original data (5.5). Evaluate how well the bootstrap method worked by assessing whether the confidence interval includes the true median and whether it provides a reasonable range for the population median.

Key Concepts

Bootstrap Sampling

Bootstrap sampling is a resampling technique used to estimate the distribution of a statistic by repeatedly sampling with replacement from the original data set. This method allows for the creation of multiple simulated samples, which can be analyzed to derive estimates such as means, medians, or confidence intervals. It is particularly useful when the sample size is small or when the underlying distribution is unknown.

Median

The median is a measure of central tendency that represents the middle value of a data set when it is ordered from least to greatest. If the data set has an odd number of observations, the median is the middle number; if even, it is the average of the two middle numbers. The median is less affected by outliers and skewed data than the mean, making it a robust measure for understanding the center of a distribution.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It is constructed using the sample data and reflects the uncertainty associated with estimating the population parameter. The endpoints of the interval, such as p2.5 and p97.5 in this context, indicate the lower and upper bounds of the interval, providing insight into the precision of the estimate.