Question

To create a confidence interval from a bootstrap distribution using percentiles, we keep the middle values and chop off some number of the lowest values and the highest values. If our bootstrap distribution contains values for 1000 bootstrap samples, indicate how many we chop off at each end for each confidence level given. (a) \(95 \%\) (b) \(90 \%\) (c) \(98 \%\) (d) \(99 \%\)

Short answer

(a) Chop off 25 values at each end for a 95% confidence interval. (b) Chop off 50 values at each end for a 90% confidence interval. (c) Chop off 10 values at each end for a 98% confidence interval. (d) Chop off 5 values at each end for a 99% confidence interval.

Step-by-step solution

Understanding bootstrap distribution and confidence intervals

A bootstrap distribution is a sample distribution of a statistic (like the mean) from a large number of resamples from a sample data set. Confidence intervals are calculated ranges within which the true population parameter value lies with a certain degree of confidence. In this case, for a given confidence level, x percent, we chop off (100 - x)/2 percent of the values at each end of the bootstrap distribution.

Chop off values for 95% confidence interval

For a 95% confidence interval, we chop off (100 - 95)/2 = 2.5% of values at each end of the bootstrap distribution. Since our bootstrap distribution contains 1000 bootstrap samples, we chop off 0.025 * 1000 = 25 values at each end.

Chop off values for 90% confidence interval

For a 90% confidence interval, we chop off (100 - 90)/2 = 5% of values at each end of the bootstrap distribution. So, we chop off 0.05 * 1000 = 50 values at each end.

Chop off values for 98% confidence interval

For a 98% confidence interval, we chop off (100 - 98)/2 = 1% of values at each end of the bootstrap distribution. So, we chop off 0.01 * 1000 = 10 values at each end.

Chop off values for 99% confidence interval

For a 99% confidence interval, we chop off (100 - 99)/2 = 0.5% of values at each end of the bootstrap distribution. So, we chop off 0.005 * 1000 = 5 values at each end.

Key concepts

Bootstrap Distribution

The bootstrap distribution is a powerful statistical tool used to estimate the sampling distribution of a statistic, such as the mean or median, from a single sample. This technique involves repeatedly resampling the original dataset with replacement to create many simulated samples, known as bootstrap samples.

These simulations generate a myriad of statistic values which form the bootstrap distribution. By analyzing this distribution, we can gain insight into the variability and possible values of the statistic in the larger population. The bootstrap distribution is particularly useful when the actual distribution of the data is unknown or when the sample size is too small for standard statistical methods to be reliable.

In educational terms, imagine you have only one bag of assorted candies to estimate the percentage of each flavor in the entire candy population. By repeatedly taking handfuls out of the bag, then putting them back and taking another handful, you start to build a picture of what the whole population might look like. Similarly, bootstrap distribution provides a model to understand the whole population from a single sample.

Percentile-Based Confidence Interval

A percentile-based confidence interval uses percentiles of the bootstrap distribution to set the bounds for the interval where the true population parameter is believed to lie with a specified level of confidence, such as 95% or 99%.

This method uses the idea that certain percentages of the bootstrap distribution fall below and above the true parameter. For example, in a 95% confidence interval, the goal is to find two points on the distribution where 2.5% of the values fall below the lower bound and 2.5% above the upper bound, leaving 95% of values in between.

Using percentile-based intervals is like cutting off the ends of a rope so that the length in-between covers a specific portion of the total length. If that rope represents all possible outcomes based on resampling, the middle part is your confidence interval where the true average length is likely to be found. This approach offers a simple, non-parametric way to estimate intervals when data doesn't necessarily follow a normal distribution or when theoretical distributions are difficult to apply.

Resampling Statistics

Resampling statistics is a branch of statistics that involves drawing repeated samples from observed data with the aim of estimating a population characteristic. This non-parametric approach doesn't rely on assumptions about the data distribution and is embodied by techniques such as bootstrapping and the permutation test.

The essence of resampling is to use the actual data to generate new samples that could represent possible outcomes were the study or experiment repeated. It's like shuffling a deck of cards several times to see the different possible sequences - each shuffle gives a potential outcome of the cards. Similarly, resampling allows statisticians to draw conclusions about the population from which the original sample was taken.

In the context of educational exercises, it's crucial for learners to understand that resampling provides a method to assess the reliability of sample statistics without relying heavily on large samples or normal distribution assumptions. It essentially enables students to make more robust inferences based on the data at hand.